Have a query? Ask a Tutor!!

Access to best educators and exclusive paper discussions

Can’t find what you’re looking for?

Our tutors are from top schools around the world, and they are waiting for you 24/7, anytime, anywhere.

Homework- One to One Solutions
Understand concepts to solve better
Get your doubts solved instantly

Maths-

General

Easy

Question

The value of $sum from n equals 1 to straight infinity of fraction numerator tan begin display style space end style open parentheses theta over 2 to the power of n close parentheses over denominator 2 to the power of n minus 1 end exponent cos space theta over 2 to the power of n minus 1 end exponent end fraction$ is

$\frac{2}{\sin 2 θ} - \frac{1}{θ}$
$\frac{2}{\sin 2 θ} + \frac{1}{θ}$
$\frac{1}{\sin 2 θ} - \frac{1}{θ}$
$\frac{1}{\sin θ} - \frac{1}{θ}$

Hint:

In this question, we have to find the value of $sum from n equals 1 to straight infinity of fraction numerator tan begin display style space end style open parentheses theta over 2 to the power of n close parentheses over denominator 2 to the power of n minus 1 end exponent cos space theta over 2 to the power of n minus 1 end exponent end fraction$ . For this first we will simplify the trigonometric function. Then as we have to find the sum of infinity terms, so, we can use the limit and so solving the sima and the limit we can get the required value.

The correct answer is: $fraction numerator 2 over denominator sin space 2 theta end fraction minus 1 over theta$

$G i v e n comma space sum from n equals 1 to straight infinity of fraction numerator tan begin display style space end style open parentheses theta over 2 to the power of n close parentheses over denominator 2 to the power of n minus 1 end exponent cos space theta over 2 to the power of n minus 1 end exponent end fraction fraction numerator tan begin display style space end style open parentheses theta over 2 to the power of n close parentheses over denominator 2 to the power of n minus 1 end exponent cos space theta over 2 to the power of n minus 1 end exponent end fraction equals fraction numerator sin begin display style theta over 2 to the power of n end style over denominator 2 to the power of n minus 1 end exponent cos begin display style theta over 2 to the power of n end style cos space begin display style theta over 2 to the power of n minus 1 end exponent end style end fraction m u l t i p l y i n g space n u m e r a t o r space a n d space d e n o m i n t a o r space b y space 2 sin theta over 2 to the power of n equals fraction numerator 2 sin squared theta over 2 to the power of n over denominator 2 to the power of n minus 1 end exponent space 2 sin theta over 2 to the power of n cos theta over 2 to the power of n cos space theta over 2 to the power of n minus 1 end exponent end fraction equals fraction numerator 1 minus cos theta over 2 to the power of n minus 1 end exponent over denominator 2 to the power of n minus 1 end exponent space sin theta over 2 to the power of n minus 1 end exponent cos space theta over 2 to the power of n minus 1 end exponent end fraction equals fraction numerator 1 minus cos theta over 2 to the power of n minus 1 end exponent over denominator 2 to the power of n minus 2 end exponent space sin theta over 2 to the power of n minus 2 end exponent end fraction equals 1 over 2 to the power of n minus 2 end exponent open parentheses fraction numerator 1 over denominator sin space theta over 2 to the power of n minus 2 end exponent end fraction minus fraction numerator cos space theta over 2 to the power of n minus 1 end exponent over denominator 2 sin space theta over 2 to the power of n minus 1 end exponent cos space theta over 2 to the power of n minus 1 end exponent end fraction close parentheses equals fraction numerator 1 over denominator 2 to the power of n minus 2 end exponent sin space theta over 2 to the power of n minus 2 end exponent end fraction minus fraction numerator 1 over denominator 2 to the power of n minus 1 end exponent sin space theta over 2 to the power of n minus 1 end exponent end fraction N o w comma space sum from n equals 1 to straight infinity of fraction numerator 1 over denominator 2 to the power of n minus 2 end exponent sin space theta over 2 to the power of n minus 2 end exponent end fraction minus fraction numerator 1 over denominator 2 to the power of n minus 1 end exponent sin space theta over 2 to the power of n minus 1 end exponent end fraction equals limit as n rightwards arrow infinity of open parentheses sum from r equals 1 to straight infinity of fraction numerator 1 over denominator 2 to the power of r minus 2 end exponent sin space theta over 2 to the power of r minus 2 end exponent end fraction minus fraction numerator 1 over denominator 2 to the power of r minus 1 end exponent sin space theta over 2 to the power of r minus 1 end exponent end fraction close parentheses equals limit as n rightwards arrow infinity of open parentheses fraction numerator 1 over denominator 2 to the power of negative 1 end exponent sin space 2 theta end fraction minus fraction numerator 1 over denominator sin space theta end fraction plus fraction numerator 1 over denominator sin space theta end fraction minus fraction numerator 1 over denominator 2 sin space begin display style bevelled theta over 2 end style end fraction plus....... fraction numerator 1 over denominator 2 to the power of n minus 2 end exponent sin space theta over 2 to the power of n minus 2 end exponent end fraction minus fraction numerator 1 over denominator 2 to the power of n minus 1 end exponent sin space theta over 2 to the power of n minus 1 end exponent end fraction close parentheses equals limit as n rightwards arrow infinity of open parentheses fraction numerator 2 over denominator sin space 2 theta end fraction minus fraction numerator 1 over denominator 2 to the power of n minus 1 end exponent sin space theta over 2 to the power of n minus 1 end exponent end fraction close parentheses equals limit as n rightwards arrow infinity of fraction numerator 2 over denominator sin space 2 theta end fraction minus limit as n rightwards arrow infinity of fraction numerator 1 over denominator 2 to the power of n minus 1 end exponent sin space theta over 2 to the power of n minus 1 end exponent end fraction equals fraction numerator 2 over denominator sin space 2 theta end fraction minus limit as n rightwards arrow infinity of fraction numerator 1 over denominator 2 to the power of n minus 1 end exponent begin display style fraction numerator sin space theta over 2 to the power of n minus 1 end exponent over denominator theta over 2 to the power of n minus 1 end exponent end fraction end style cross times theta over 2 to the power of n minus 1 end exponent end fraction equals fraction numerator 2 over denominator sin space 2 theta end fraction minus limit as n rightwards arrow infinity of 1 over theta equals fraction numerator 2 over denominator sin space 2 theta end fraction minus 1 over theta$

Related Questions to study

If the equation $cot to the power of 4 space x minus 2 cos space e c squared x plus a squared$ =0 has at least one solution, then the sum of all possible integral values of 'a' is equal to

G i v e n comma space cot to the power of 4 space x minus 2 cos e c squared x plus a squared equals 0 rightwards double arrow c o t to the power of 4 x minus 2 open parentheses 1 plus c o t squared x close parentheses plus a squared equals 0 rightwards double arrow c o t to the power of 4 x minus 2 c o t squared x plus a squared minus 2 equals 0 rightwards double arrow c o t squared x equals fraction numerator 2 plus-or-minus square root of 4 minus 4 open parentheses a squared minus 2 close parentheses end root over denominator 2 end fraction f o r space r e a l space r o o t comma space 4 minus 4 open parentheses a squared minus 2 close parentheses greater than 0 1 minus open parentheses a squared minus 2 close parentheses greater than 0 1 minus a squared plus 2 greater than o 3 minus a squared greater than 0 3 greater than a squared plus-or-minus square root of 3 greater than a t h e r e f o r e comma a element of open parentheses negative square root of 3 comma square root of 3 close parentheses S o comma space s u m space o f space v a l u e s space o f space a equals 0.

If the equation $cot to the power of 4 space x minus 2 cos space e c squared x plus a squared$ =0 has at least one solution, then the sum of all possible integral values of 'a' is equal to

G i v e n comma space cot to the power of 4 space x minus 2 cos e c squared x plus a squared equals 0 rightwards double arrow c o t to the power of 4 x minus 2 open parentheses 1 plus c o t squared x close parentheses plus a squared equals 0 rightwards double arrow c o t to the power of 4 x minus 2 c o t squared x plus a squared minus 2 equals 0 rightwards double arrow c o t squared x equals fraction numerator 2 plus-or-minus square root of 4 minus 4 open parentheses a squared minus 2 close parentheses end root over denominator 2 end fraction f o r space r e a l space r o o t comma space 4 minus 4 open parentheses a squared minus 2 close parentheses greater than 0 1 minus open parentheses a squared minus 2 close parentheses greater than 0 1 minus a squared plus 2 greater than o 3 minus a squared greater than 0 3 greater than a squared plus-or-minus square root of 3 greater than a t h e r e f o r e comma a element of open parentheses negative square root of 3 comma square root of 3 close parentheses S o comma space s u m space o f space v a l u e s space o f space a equals 0.

$cos to the power of 6 space pi over 16 plus cos to the power of 6 space fraction numerator 3 pi over denominator 16 end fraction plus cos to the power of 6 space fraction numerator 5 pi over denominator 16 end fraction plus cos to the power of 6 space fraction numerator 7 pi over denominator 16 end fraction$ =

cos to the power of 6 space pi over 16 plus cos to the power of 6 space fraction numerator 3 pi over denominator 16 end fraction plus cos to the power of 6 space fraction numerator 5 pi over denominator 16 end fraction plus cos to the power of 6 space fraction numerator 7 pi over denominator 16 end fraction equals cos to the power of 6 straight pi over 16 plus cos to the power of 6 open parentheses straight pi over 2 minus straight pi over 16 close parentheses plus cos to the power of 6 fraction numerator 3 straight pi over denominator 16 end fraction plus cos to the power of 6 open parentheses straight pi over 2 minus fraction numerator 3 straight pi over denominator 16 end fraction close parentheses equals cos to the power of 6 straight pi over 16 plus sin to the power of 6 straight pi over 16 plus cos to the power of 6 fraction numerator 3 straight pi over denominator 16 end fraction plus sin to the power of 6 fraction numerator 3 straight pi over denominator 16 end fraction space space space space space space space space space space space space space space space open square brackets cos open parentheses straight pi over 2 minus A close parentheses equals sin A close square brackets equals open parentheses sin squared straight pi over 16 plus cos squared straight pi over 16 close parentheses open parentheses sin to the power of 4 straight pi over 16 plus cos to the power of 4 straight pi over 16 minus sin squared straight pi over 16 cos squared straight pi over 16 close parentheses plus open parentheses sin squared fraction numerator 3 straight pi over denominator 16 end fraction plus cos squared fraction numerator 3 straight pi over denominator 16 end fraction close parentheses open parentheses sin to the power of 4 fraction numerator 3 straight pi over denominator 16 end fraction plus cos to the power of 4 fraction numerator 3 straight pi over denominator 16 end fraction minus sin squared fraction numerator 3 straight pi over denominator 16 end fraction cos squared fraction numerator 3 straight pi over denominator 16 end fraction close parentheses space space space space open square brackets a cubed plus b cubed equals open parentheses a plus b close parentheses open parentheses a squared plus b squared minus a b close parentheses close square brackets equals open square brackets open parentheses sin squared straight pi over 16 plus cos squared straight pi over 16 close parentheses squared minus 2 sin squared straight pi over 16 cos squared straight pi over 16 minus sin squared straight pi over 16 cos squared straight pi over 16 close square brackets plus open square brackets open parentheses sin squared fraction numerator 3 straight pi over denominator 16 end fraction plus cos squared fraction numerator 3 straight pi over denominator 16 end fraction close parentheses squared minus 2 sin squared fraction numerator 3 straight pi over denominator 16 end fraction cos squared fraction numerator 3 straight pi over denominator 16 end fraction minus sin squared fraction numerator 3 straight pi over denominator 16 end fraction cos squared fraction numerator 3 straight pi over denominator 16 end fraction close square brackets space space space space space space space space open square brackets a squared plus b squared equals open parentheses a plus b close parentheses squared minus 2 a b close square brackets equals open parentheses 1 minus 3 sin squared straight pi over 16 cos squared straight pi over 16 close parentheses plus open parentheses 1 minus 3 sin squared fraction numerator 3 straight pi over denominator 16 end fraction cos squared fraction numerator 3 straight pi over denominator 16 end fraction close parentheses equals 1 minus 3 over 4 sin squared straight pi over 8 plus 1 minus 3 over 4 sin squared fraction numerator 3 straight pi over denominator 8 end fraction space space space space space space space open square brackets 4 sin squared A cos squared A equals sin squared 2 A close square brackets equals 2 minus 3 over 4 open square brackets sin squared straight pi over 8 plus sin squared fraction numerator 3 straight pi over denominator 8 end fraction close square brackets equals 2 minus 3 over 4 open square brackets sin squared straight pi over 8 plus sin squared open parentheses straight pi over 2 minus straight pi over 8 close parentheses close square brackets equals 2 minus 3 over 4 open square brackets sin squared straight pi over 8 plus cos squared straight pi over 8 close square brackets space space space space space space space space space space space space space space space open square brackets sin open parentheses straight pi over 2 minus A close parentheses equals cos A close square brackets equals 2 minus 3 over 4 equals 5 over 4

$cos to the power of 6 space pi over 16 plus cos to the power of 6 space fraction numerator 3 pi over denominator 16 end fraction plus cos to the power of 6 space fraction numerator 5 pi over denominator 16 end fraction plus cos to the power of 6 space fraction numerator 7 pi over denominator 16 end fraction$ =

cos to the power of 6 space pi over 16 plus cos to the power of 6 space fraction numerator 3 pi over denominator 16 end fraction plus cos to the power of 6 space fraction numerator 5 pi over denominator 16 end fraction plus cos to the power of 6 space fraction numerator 7 pi over denominator 16 end fraction equals cos to the power of 6 straight pi over 16 plus cos to the power of 6 open parentheses straight pi over 2 minus straight pi over 16 close parentheses plus cos to the power of 6 fraction numerator 3 straight pi over denominator 16 end fraction plus cos to the power of 6 open parentheses straight pi over 2 minus fraction numerator 3 straight pi over denominator 16 end fraction close parentheses equals cos to the power of 6 straight pi over 16 plus sin to the power of 6 straight pi over 16 plus cos to the power of 6 fraction numerator 3 straight pi over denominator 16 end fraction plus sin to the power of 6 fraction numerator 3 straight pi over denominator 16 end fraction space space space space space space space space space space space space space space space open square brackets cos open parentheses straight pi over 2 minus A close parentheses equals sin A close square brackets equals open parentheses sin squared straight pi over 16 plus cos squared straight pi over 16 close parentheses open parentheses sin to the power of 4 straight pi over 16 plus cos to the power of 4 straight pi over 16 minus sin squared straight pi over 16 cos squared straight pi over 16 close parentheses plus open parentheses sin squared fraction numerator 3 straight pi over denominator 16 end fraction plus cos squared fraction numerator 3 straight pi over denominator 16 end fraction close parentheses open parentheses sin to the power of 4 fraction numerator 3 straight pi over denominator 16 end fraction plus cos to the power of 4 fraction numerator 3 straight pi over denominator 16 end fraction minus sin squared fraction numerator 3 straight pi over denominator 16 end fraction cos squared fraction numerator 3 straight pi over denominator 16 end fraction close parentheses space space space space open square brackets a cubed plus b cubed equals open parentheses a plus b close parentheses open parentheses a squared plus b squared minus a b close parentheses close square brackets equals open square brackets open parentheses sin squared straight pi over 16 plus cos squared straight pi over 16 close parentheses squared minus 2 sin squared straight pi over 16 cos squared straight pi over 16 minus sin squared straight pi over 16 cos squared straight pi over 16 close square brackets plus open square brackets open parentheses sin squared fraction numerator 3 straight pi over denominator 16 end fraction plus cos squared fraction numerator 3 straight pi over denominator 16 end fraction close parentheses squared minus 2 sin squared fraction numerator 3 straight pi over denominator 16 end fraction cos squared fraction numerator 3 straight pi over denominator 16 end fraction minus sin squared fraction numerator 3 straight pi over denominator 16 end fraction cos squared fraction numerator 3 straight pi over denominator 16 end fraction close square brackets space space space space space space space space open square brackets a squared plus b squared equals open parentheses a plus b close parentheses squared minus 2 a b close square brackets equals open parentheses 1 minus 3 sin squared straight pi over 16 cos squared straight pi over 16 close parentheses plus open parentheses 1 minus 3 sin squared fraction numerator 3 straight pi over denominator 16 end fraction cos squared fraction numerator 3 straight pi over denominator 16 end fraction close parentheses equals 1 minus 3 over 4 sin squared straight pi over 8 plus 1 minus 3 over 4 sin squared fraction numerator 3 straight pi over denominator 8 end fraction space space space space space space space open square brackets 4 sin squared A cos squared A equals sin squared 2 A close square brackets equals 2 minus 3 over 4 open square brackets sin squared straight pi over 8 plus sin squared fraction numerator 3 straight pi over denominator 8 end fraction close square brackets equals 2 minus 3 over 4 open square brackets sin squared straight pi over 8 plus sin squared open parentheses straight pi over 2 minus straight pi over 8 close parentheses close square brackets equals 2 minus 3 over 4 open square brackets sin squared straight pi over 8 plus cos squared straight pi over 8 close square brackets space space space space space space space space space space space space space space space open square brackets sin open parentheses straight pi over 2 minus A close parentheses equals cos A close square brackets equals 2 minus 3 over 4 equals 5 over 4

If $a sin space x plus b cos space left parenthesis x plus theta right parenthesis plus b cos space left parenthesis x minus theta right parenthesis equals$ d then the minimum value of $vertical line cos space theta vertical line$ is equal to (where is variable)

a sin space x plus b cos space left parenthesis x plus theta right parenthesis plus b cos space left parenthesis x minus theta right parenthesis equals d rightwards double arrow a sin x plus b open parentheses cos x space cos theta minus sin x space sin theta close parentheses plus b open parentheses cos x space cos plus sin x space sin theta close parentheses equals d rightwards double arrow a space sin x plus b cos x space cos theta minus b sin x space sin theta plus b cos x space cos plus b sin x space sin theta equals d rightwards double arrow a sin x space plus 2 b. cos x. cos theta equals d rightwards double arrow a open parentheses fraction numerator 2 tan begin display style x over 2 end style over denominator 1 plus tan squared begin display style x over 2 end style end fraction close parentheses plus 2 b. open parentheses fraction numerator 1 minus tan squared begin display style x over 2 end style over denominator 1 plus tan squared begin display style x over 2 end style end fraction close parentheses. cos theta equals d rightwards double arrow 2 a space tan x over 2 plus 2 b open parentheses 1 minus tan squared x over 2 close parentheses. cos theta equals d open parentheses 1 plus tan squared x over 2 close parentheses rightwards double arrow 2 a. tan x over 2 plus 2 b cos theta minus 2 b space tan squared x over 2 cos theta minus d minus d tan squared x over 2 equals 0 rightwards double arrow negative tan squared x over 2 open parentheses 2 b cos theta plus d close parentheses plus 2 a. tan x over 2 plus 2 b cos theta minus d equals 0 rightwards double arrow tan squared x over 2 open parentheses 2 b cos theta plus d close parentheses minus 2 a. tan x over 2 minus 2 b cos theta plus d equals 0 w e space k n o w space t h a t space space D greater or equal than 0 comma b squared minus 4 a c greater or equal than 0 open parentheses negative 2 a close parentheses squared minus 4 open parentheses 2 b cos theta plus d close parentheses open parentheses d minus 2 b cos theta close parentheses greater or equal than 0 4 a squared minus 4 open square brackets d squared minus open parentheses 2 b cos theta close parentheses squared close square brackets greater or equal than 0 4 open parentheses a squared minus d squared plus 4 b squared cos squared theta close parentheses greater or equal than 0 4 b squared cos squared theta greater or equal than d squared minus a squared cos squared theta greater or equal than fraction numerator d squared minus a squared over denominator 4 b squared end fraction open vertical bar cos theta close vertical bar greater or equal than square root of fraction numerator d squared minus a squared over denominator 4 b squared end fraction end root open vertical bar cos theta close vertical bar greater or equal than fraction numerator square root of d squared minus a squared end root over denominator 2 open vertical bar b close vertical bar end fraction S o comma space t h e space m i n i m u m space v a l u e space o f open vertical bar cos theta close vertical bar equals fraction numerator square root of d squared minus a squared end root over denominator 2 open vertical bar b close vertical bar end fraction

If $a sin space x plus b cos space left parenthesis x plus theta right parenthesis plus b cos space left parenthesis x minus theta right parenthesis equals$ d then the minimum value of $vertical line cos space theta vertical line$ is equal to (where is variable)

a sin space x plus b cos space left parenthesis x plus theta right parenthesis plus b cos space left parenthesis x minus theta right parenthesis equals d rightwards double arrow a sin x plus b open parentheses cos x space cos theta minus sin x space sin theta close parentheses plus b open parentheses cos x space cos plus sin x space sin theta close parentheses equals d rightwards double arrow a space sin x plus b cos x space cos theta minus b sin x space sin theta plus b cos x space cos plus b sin x space sin theta equals d rightwards double arrow a sin x space plus 2 b. cos x. cos theta equals d rightwards double arrow a open parentheses fraction numerator 2 tan begin display style x over 2 end style over denominator 1 plus tan squared begin display style x over 2 end style end fraction close parentheses plus 2 b. open parentheses fraction numerator 1 minus tan squared begin display style x over 2 end style over denominator 1 plus tan squared begin display style x over 2 end style end fraction close parentheses. cos theta equals d rightwards double arrow 2 a space tan x over 2 plus 2 b open parentheses 1 minus tan squared x over 2 close parentheses. cos theta equals d open parentheses 1 plus tan squared x over 2 close parentheses rightwards double arrow 2 a. tan x over 2 plus 2 b cos theta minus 2 b space tan squared x over 2 cos theta minus d minus d tan squared x over 2 equals 0 rightwards double arrow negative tan squared x over 2 open parentheses 2 b cos theta plus d close parentheses plus 2 a. tan x over 2 plus 2 b cos theta minus d equals 0 rightwards double arrow tan squared x over 2 open parentheses 2 b cos theta plus d close parentheses minus 2 a. tan x over 2 minus 2 b cos theta plus d equals 0 w e space k n o w space t h a t space space D greater or equal than 0 comma b squared minus 4 a c greater or equal than 0 open parentheses negative 2 a close parentheses squared minus 4 open parentheses 2 b cos theta plus d close parentheses open parentheses d minus 2 b cos theta close parentheses greater or equal than 0 4 a squared minus 4 open square brackets d squared minus open parentheses 2 b cos theta close parentheses squared close square brackets greater or equal than 0 4 open parentheses a squared minus d squared plus 4 b squared cos squared theta close parentheses greater or equal than 0 4 b squared cos squared theta greater or equal than d squared minus a squared cos squared theta greater or equal than fraction numerator d squared minus a squared over denominator 4 b squared end fraction open vertical bar cos theta close vertical bar greater or equal than square root of fraction numerator d squared minus a squared over denominator 4 b squared end fraction end root open vertical bar cos theta close vertical bar greater or equal than fraction numerator square root of d squared minus a squared end root over denominator 2 open vertical bar b close vertical bar end fraction S o comma space t h e space m i n i m u m space v a l u e space o f open vertical bar cos theta close vertical bar equals fraction numerator square root of d squared minus a squared end root over denominator 2 open vertical bar b close vertical bar end fraction

parallel

$a squared plus b squared plus 2 a b cos space theta equals 1 comma c squared plus d squared plus 2 c d cos space theta equals 1$ and $a c plus b d plus left parenthesis a d plus b c right parenthesis cos space theta equals 0$ then $a squared plus c squared$ =

$a squared plus b squared plus 2 a b cos space theta equals 1 comma c squared plus d squared plus 2 c d cos space theta equals 1$ and $a c plus b d plus left parenthesis a d plus b c right parenthesis cos space theta equals 0$ then $a squared plus c squared$ =

$sin to the power of 4 space pi over 16 plus sin to the power of 4 fraction numerator 3 pi over denominator 16 end fraction plus sin to the power of 4 space fraction numerator 5 pi over denominator 16 end fraction plus sin to the power of 4 space fraction numerator 7 pi over denominator 16 end fraction$

sin to the power of 4 space pi over 16 plus sin to the power of 4 fraction numerator 3 pi over denominator 16 end fraction plus sin to the power of 4 space fraction numerator 5 pi over denominator 16 end fraction plus sin to the power of 4 space fraction numerator 7 pi over denominator 16 end fraction equals sin to the power of 4 straight pi over 16 plus sin to the power of 4 open parentheses straight pi over 2 minus straight pi over 16 close parentheses plus sin to the power of 4 fraction numerator 3 straight pi over denominator 16 end fraction plus sin to the power of 4 open parentheses straight pi over 2 minus fraction numerator 3 straight pi over denominator 16 end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals sin to the power of 4 straight pi over 16 plus cos to the power of 4 straight pi over 16 plus sin to the power of 4 fraction numerator 3 straight pi over denominator 16 end fraction plus cos to the power of 4 fraction numerator 3 straight pi over denominator 16 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets sin open parentheses straight pi over 2 minus A close parentheses equals cos A close square brackets equals open parentheses sin squared straight pi over 16 plus cos squared straight pi over 16 close parentheses squared minus 2 sin squared straight pi over 16 cos squared straight pi over 16 plus open parentheses sin squared fraction numerator 3 straight pi over denominator 16 end fraction plus cos squared fraction numerator 3 straight pi over denominator 16 end fraction close parentheses squared minus 2 sin squared fraction numerator 3 straight pi over denominator 16 end fraction cos squared fraction numerator 3 straight pi over denominator 16 end fraction space space space space space equals 1 minus 1 half sin squared straight pi over 8 plus 1 minus 1 half sin squared fraction numerator 3 straight pi over denominator 8 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets 4 sin squared A. cos squared A equals sin squared 2 A close square brackets equals 2 minus 1 half open square brackets sin squared straight pi over 8 plus sin squared fraction numerator 3 straight pi over denominator 8 end fraction close square brackets equals 2 minus 1 half open square brackets sin squared straight pi over 8 plus sin squared open parentheses straight pi over 2 minus straight pi over 8 close parentheses close square brackets equals 2 minus 1 half open square brackets sin squared straight pi over 8 plus cos squared straight pi over 8 close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets sin open parentheses straight pi over 2 minus A close parentheses equals cos A close square brackets equals 2 minus 1 half equals 3 over 2

$sin to the power of 4 space pi over 16 plus sin to the power of 4 fraction numerator 3 pi over denominator 16 end fraction plus sin to the power of 4 space fraction numerator 5 pi over denominator 16 end fraction plus sin to the power of 4 space fraction numerator 7 pi over denominator 16 end fraction$

sin to the power of 4 space pi over 16 plus sin to the power of 4 fraction numerator 3 pi over denominator 16 end fraction plus sin to the power of 4 space fraction numerator 5 pi over denominator 16 end fraction plus sin to the power of 4 space fraction numerator 7 pi over denominator 16 end fraction equals sin to the power of 4 straight pi over 16 plus sin to the power of 4 open parentheses straight pi over 2 minus straight pi over 16 close parentheses plus sin to the power of 4 fraction numerator 3 straight pi over denominator 16 end fraction plus sin to the power of 4 open parentheses straight pi over 2 minus fraction numerator 3 straight pi over denominator 16 end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals sin to the power of 4 straight pi over 16 plus cos to the power of 4 straight pi over 16 plus sin to the power of 4 fraction numerator 3 straight pi over denominator 16 end fraction plus cos to the power of 4 fraction numerator 3 straight pi over denominator 16 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets sin open parentheses straight pi over 2 minus A close parentheses equals cos A close square brackets equals open parentheses sin squared straight pi over 16 plus cos squared straight pi over 16 close parentheses squared minus 2 sin squared straight pi over 16 cos squared straight pi over 16 plus open parentheses sin squared fraction numerator 3 straight pi over denominator 16 end fraction plus cos squared fraction numerator 3 straight pi over denominator 16 end fraction close parentheses squared minus 2 sin squared fraction numerator 3 straight pi over denominator 16 end fraction cos squared fraction numerator 3 straight pi over denominator 16 end fraction space space space space space equals 1 minus 1 half sin squared straight pi over 8 plus 1 minus 1 half sin squared fraction numerator 3 straight pi over denominator 8 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets 4 sin squared A. cos squared A equals sin squared 2 A close square brackets equals 2 minus 1 half open square brackets sin squared straight pi over 8 plus sin squared fraction numerator 3 straight pi over denominator 8 end fraction close square brackets equals 2 minus 1 half open square brackets sin squared straight pi over 8 plus sin squared open parentheses straight pi over 2 minus straight pi over 8 close parentheses close square brackets equals 2 minus 1 half open square brackets sin squared straight pi over 8 plus cos squared straight pi over 8 close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets sin open parentheses straight pi over 2 minus A close parentheses equals cos A close square brackets equals 2 minus 1 half equals 3 over 2

If $cos space left parenthesis A plus B plus C right parenthesis equals cos space A cos space B cos space C comma$ then $fraction numerator 8 sin space left parenthesis B plus C right parenthesis sin space left parenthesis C plus A right parenthesis sin space left parenthesis A plus B right parenthesis over denominator sin space 2 A sin space 2 B sin space 2 C end fraction$ =

cos space left parenthesis A plus B plus C right parenthesis equals cos space A cos space B cos space C rightwards double arrow cos A cos open parentheses B plus C close parentheses equals cos A. cos B. cos C rightwards double arrow cos A. cos open parentheses B plus C close parentheses minus sin A sin open parentheses B plus C close parentheses equals cos A. cos B. cos C rightwards double arrow cos A. cos open parentheses B plus C close parentheses minus cos A. cos B. cos C equals sin A sin open parentheses B plus C close parentheses rightwards double arrow cos A open square brackets cos open parentheses B plus C close parentheses minus cos B. cos C close square brackets equals sin A. sin open parentheses B plus C close parentheses rightwards double arrow cos A open square brackets cos B. cos C minus sin B. sin C minus cos B. cos C close square brackets equals sin A. sin open parentheses B plus C close parentheses rightwards double arrow sin open parentheses B plus C close parentheses equals fraction numerator negative cos A. sin B. sin C over denominator sin A end fraction S i m i l a r l y comma space sin open parentheses C plus A close parentheses equals fraction numerator negative cos B. sin C. sin A over denominator sin B end fraction sin open parentheses A plus B close parentheses equals fraction numerator negative cos C. sin A. sin B over denominator sin C end fraction sin open parentheses A plus B close parentheses. sin open parentheses B plus C close parentheses. sin open parentheses A plus C close parentheses equals fraction numerator negative cos A. sin B. sin C over denominator sin A end fraction. fraction numerator negative cos B. sin C. sin A over denominator sin B end fraction. fraction numerator negative cos C. sin A. sin B over denominator sin C end fraction rightwards double arrow sin open parentheses A plus B close parentheses. sin open parentheses B plus C close parentheses. sin open parentheses A plus C close parentheses equals negative sin A. sin B. sin C. cos A. cos B. cos C rightwards double arrow sin open parentheses A plus B close parentheses. sin open parentheses B plus C close parentheses. sin open parentheses A plus C close parentheses equals fraction numerator negative sin 2 A. sin 2 B. sin 2 C over denominator 8 end fraction rightwards double arrow fraction numerator 8 sin open parentheses A plus B close parentheses. sin open parentheses B plus C close parentheses. sin open parentheses A plus C close parentheses over denominator sin 2 A. sin 2 B. sin 2 C end fraction equals negative 1

If $cos space left parenthesis A plus B plus C right parenthesis equals cos space A cos space B cos space C comma$ then $fraction numerator 8 sin space left parenthesis B plus C right parenthesis sin space left parenthesis C plus A right parenthesis sin space left parenthesis A plus B right parenthesis over denominator sin space 2 A sin space 2 B sin space 2 C end fraction$ =

cos space left parenthesis A plus B plus C right parenthesis equals cos space A cos space B cos space C rightwards double arrow cos A cos open parentheses B plus C close parentheses equals cos A. cos B. cos C rightwards double arrow cos A. cos open parentheses B plus C close parentheses minus sin A sin open parentheses B plus C close parentheses equals cos A. cos B. cos C rightwards double arrow cos A. cos open parentheses B plus C close parentheses minus cos A. cos B. cos C equals sin A sin open parentheses B plus C close parentheses rightwards double arrow cos A open square brackets cos open parentheses B plus C close parentheses minus cos B. cos C close square brackets equals sin A. sin open parentheses B plus C close parentheses rightwards double arrow cos A open square brackets cos B. cos C minus sin B. sin C minus cos B. cos C close square brackets equals sin A. sin open parentheses B plus C close parentheses rightwards double arrow sin open parentheses B plus C close parentheses equals fraction numerator negative cos A. sin B. sin C over denominator sin A end fraction S i m i l a r l y comma space sin open parentheses C plus A close parentheses equals fraction numerator negative cos B. sin C. sin A over denominator sin B end fraction sin open parentheses A plus B close parentheses equals fraction numerator negative cos C. sin A. sin B over denominator sin C end fraction sin open parentheses A plus B close parentheses. sin open parentheses B plus C close parentheses. sin open parentheses A plus C close parentheses equals fraction numerator negative cos A. sin B. sin C over denominator sin A end fraction. fraction numerator negative cos B. sin C. sin A over denominator sin B end fraction. fraction numerator negative cos C. sin A. sin B over denominator sin C end fraction rightwards double arrow sin open parentheses A plus B close parentheses. sin open parentheses B plus C close parentheses. sin open parentheses A plus C close parentheses equals negative sin A. sin B. sin C. cos A. cos B. cos C rightwards double arrow sin open parentheses A plus B close parentheses. sin open parentheses B plus C close parentheses. sin open parentheses A plus C close parentheses equals fraction numerator negative sin 2 A. sin 2 B. sin 2 C over denominator 8 end fraction rightwards double arrow fraction numerator 8 sin open parentheses A plus B close parentheses. sin open parentheses B plus C close parentheses. sin open parentheses A plus C close parentheses over denominator sin 2 A. sin 2 B. sin 2 C end fraction equals negative 1

parallel

The maximum value of the expression $open vertical bar square root of open parentheses sin squared space x plus 2 a squared close parentheses end root minus square root of open parentheses 2 a squared minus 1 minus cos squared space x close parentheses end root close vertical bar$
where $a$ and x are real numbers is

open vertical bar square root of open parentheses sin squared space x plus 2 a squared close parentheses end root minus square root of open parentheses 2 a squared minus 1 minus cos squared space x close parentheses end root close vertical bar equals open vertical bar square root of open parentheses sin squared space x plus 2 a squared close parentheses end root minus square root of open parentheses 2 a squared minus 1 minus 1 minus sin squared space x close parentheses end root close vertical bar equals open vertical bar square root of open parentheses sin squared space x plus 2 a squared close parentheses end root minus square root of open parentheses sin squared space x plus 2 a squared minus 2 close parentheses end root close vertical bar l e t space sin squared x plus 2 a squared space b e space y. equals open vertical bar square root of y minus square root of y minus 2 end root close vertical bar S i n c e space b o t h space t h e space t e r m s space a r e space p o s i t i v e comma space t h e space g i v e n space f u n c t i o n space w i l l space h a v e space t h e space m a x i m u m space v a l u e space w h e n space y equals 2 i. e. comma space square root of y minus 2 end root equals 0 N o w comma space open vertical bar square root of y minus square root of y minus 2 end root close vertical bar equals square root of 2

The maximum value of the expression $open vertical bar square root of open parentheses sin squared space x plus 2 a squared close parentheses end root minus square root of open parentheses 2 a squared minus 1 minus cos squared space x close parentheses end root close vertical bar$
where $a$ and x are real numbers is

open vertical bar square root of open parentheses sin squared space x plus 2 a squared close parentheses end root minus square root of open parentheses 2 a squared minus 1 minus cos squared space x close parentheses end root close vertical bar equals open vertical bar square root of open parentheses sin squared space x plus 2 a squared close parentheses end root minus square root of open parentheses 2 a squared minus 1 minus 1 minus sin squared space x close parentheses end root close vertical bar equals open vertical bar square root of open parentheses sin squared space x plus 2 a squared close parentheses end root minus square root of open parentheses sin squared space x plus 2 a squared minus 2 close parentheses end root close vertical bar l e t space sin squared x plus 2 a squared space b e space y. equals open vertical bar square root of y minus square root of y minus 2 end root close vertical bar S i n c e space b o t h space t h e space t e r m s space a r e space p o s i t i v e comma space t h e space g i v e n space f u n c t i o n space w i l l space h a v e space t h e space m a x i m u m space v a l u e space w h e n space y equals 2 i. e. comma space square root of y minus 2 end root equals 0 N o w comma space open vertical bar square root of y minus square root of y minus 2 end root close vertical bar equals square root of 2

If $tan space theta minus cot space theta equals a$ and $cos space theta minus sin space theta equals b$ then $open parentheses a squared plus 4 close parentheses open parentheses b squared minus 1 close parentheses squared$

tan space theta minus cot space theta equals a rightwards double arrow fraction numerator sin theta over denominator cos theta end fraction minus fraction numerator cos theta over denominator sin theta end fraction equals a space space space space space space space rightwards double arrow fraction numerator sin squared theta minus cos squared theta over denominator cos theta sin theta end fraction equals a rightwards double arrow fraction numerator negative 2 cos 2 theta over denominator sin 2 theta end fraction equals a space space space space space space space space space open square brackets cos squared x minus sin squared x equals cos 2 x comma space 2 sin x space cos x space equals 2 sin 2 x close square brackets s q u a r i n g space b o t h space s i d e s comma rightwards double arrow fraction numerator 4 cos squared 2 theta over denominator sin squared 2 theta end fraction equals a squared a d d i n g space 4 space o n space b o t h space s i d e s comma rightwards double arrow fraction numerator 4 cos squared 2 theta over denominator sin squared 2 theta end fraction plus 4 equals a squared plus 4 rightwards double arrow fraction numerator 4 cos squared 2 theta plus 4 sin squared 2 theta over denominator sin squared 2 theta end fraction equals a squared plus 4 rightwards double arrow fraction numerator 4 over denominator sin squared 2 theta end fraction equals a squared plus 4........ e q u a t i o n space 1 space space space space space space space space space space space space space space open square brackets cos squared x plus sin squared x equals 1 close square brackets

cos space theta space minus sin theta space equals b s q u a r i n g space b o t h space s i d e s comma space rightwards double arrow open parentheses cos theta minus sin theta close parentheses squared equals b squared rightwards double arrow cos squared theta plus sin squared theta minus 2 sin theta cos theta equals b squared rightwards double arrow 1 minus sin 2 theta equals b squared space space space space space space open square brackets cos squared theta plus sin squared theta equals 1 comma space 2 sin theta cos theta equals sin 2 theta close square brackets rightwards double arrow negative sin 2 theta equals b squared minus 1 s q u a r i n g space b o t h space s i d e s comma rightwards double arrow sin squared 2 theta equals open parentheses b squared minus 1 close parentheses squared......... e q u a t i o n space 2

on multiplying equation 1 and 2,

open parentheses a squared plus 4 close parentheses open parentheses b squared minus 1 close parentheses squared equals fraction numerator 4 over denominator sin squared 2 theta end fraction cross times sin squared 2 theta equals 4

If $tan space theta minus cot space theta equals a$ and $cos space theta minus sin space theta equals b$ then $open parentheses a squared plus 4 close parentheses open parentheses b squared minus 1 close parentheses squared$

tan space theta minus cot space theta equals a rightwards double arrow fraction numerator sin theta over denominator cos theta end fraction minus fraction numerator cos theta over denominator sin theta end fraction equals a space space space space space space space rightwards double arrow fraction numerator sin squared theta minus cos squared theta over denominator cos theta sin theta end fraction equals a rightwards double arrow fraction numerator negative 2 cos 2 theta over denominator sin 2 theta end fraction equals a space space space space space space space space space open square brackets cos squared x minus sin squared x equals cos 2 x comma space 2 sin x space cos x space equals 2 sin 2 x close square brackets s q u a r i n g space b o t h space s i d e s comma rightwards double arrow fraction numerator 4 cos squared 2 theta over denominator sin squared 2 theta end fraction equals a squared a d d i n g space 4 space o n space b o t h space s i d e s comma rightwards double arrow fraction numerator 4 cos squared 2 theta over denominator sin squared 2 theta end fraction plus 4 equals a squared plus 4 rightwards double arrow fraction numerator 4 cos squared 2 theta plus 4 sin squared 2 theta over denominator sin squared 2 theta end fraction equals a squared plus 4 rightwards double arrow fraction numerator 4 over denominator sin squared 2 theta end fraction equals a squared plus 4........ e q u a t i o n space 1 space space space space space space space space space space space space space space open square brackets cos squared x plus sin squared x equals 1 close square brackets

cos space theta space minus sin theta space equals b s q u a r i n g space b o t h space s i d e s comma space rightwards double arrow open parentheses cos theta minus sin theta close parentheses squared equals b squared rightwards double arrow cos squared theta plus sin squared theta minus 2 sin theta cos theta equals b squared rightwards double arrow 1 minus sin 2 theta equals b squared space space space space space space open square brackets cos squared theta plus sin squared theta equals 1 comma space 2 sin theta cos theta equals sin 2 theta close square brackets rightwards double arrow negative sin 2 theta equals b squared minus 1 s q u a r i n g space b o t h space s i d e s comma rightwards double arrow sin squared 2 theta equals open parentheses b squared minus 1 close parentheses squared......... e q u a t i o n space 2

on multiplying equation 1 and 2,

open parentheses a squared plus 4 close parentheses open parentheses b squared minus 1 close parentheses squared equals fraction numerator 4 over denominator sin squared 2 theta end fraction cross times sin squared 2 theta equals 4

The maximum value of $open parentheses cos space alpha subscript 1 close parentheses open parentheses cos space alpha subscript 2 close parentheses horizontal ellipsis open parentheses cos space alpha subscript n close parentheses$ under the restrictions $0 less or equal than alpha subscript 1 comma alpha subscript 2 horizontal ellipsis horizontal ellipsis alpha subscript n less or equal than pi over 2$ and $cot space alpha subscript 1 times cot space alpha subscript 2 horizontal ellipsis horizontal ellipsis cot space alpha subscript n equals 1$ is

cot space alpha subscript 1 times cot space alpha subscript 2 horizontal ellipsis horizontal ellipsis cot space alpha subscript n equals 1

rightwards double arrow fraction numerator cos alpha subscript 1 over denominator sin alpha subscript 1 end fraction space. fraction numerator cos alpha subscript 2 over denominator sin alpha subscript 2 end fraction space. fraction numerator cos alpha subscript 3 over denominator sin alpha subscript 3 end fraction..... fraction numerator cos alpha subscript n over denominator sin alpha subscript n end fraction equals 1 rightwards double arrow cos alpha subscript 1. space cos alpha subscript 2. space cos alpha subscript 3...... cos alpha subscript n equals sin alpha subscript 1. space sin alpha subscript 2. space sin alpha subscript 3...... sin alpha subscript n t h i s space i s space p o s s i b l e space o n l y space w h e n space alpha subscript 1 equals alpha subscript 2 equals space alpha subscript 3 equals..... alpha subscript n equals 45 degree S o comma space cos alpha subscript 1. space cos alpha subscript 2. space cos alpha subscript 3...... cos alpha subscript n equals fraction numerator 1 over denominator square root of 2 end fraction. fraction numerator 1 over denominator square root of 2 end fraction. fraction numerator 1 over denominator square root of 2 end fraction. fraction numerator 1 over denominator square root of 2 end fraction....... fraction numerator 1 over denominator square root of 2 end fraction equals 1 over 2 to the power of begin display style n over 2 end style end exponent

The maximum value of $open parentheses cos space alpha subscript 1 close parentheses open parentheses cos space alpha subscript 2 close parentheses horizontal ellipsis open parentheses cos space alpha subscript n close parentheses$ under the restrictions $0 less or equal than alpha subscript 1 comma alpha subscript 2 horizontal ellipsis horizontal ellipsis alpha subscript n less or equal than pi over 2$ and $cot space alpha subscript 1 times cot space alpha subscript 2 horizontal ellipsis horizontal ellipsis cot space alpha subscript n equals 1$ is

cot space alpha subscript 1 times cot space alpha subscript 2 horizontal ellipsis horizontal ellipsis cot space alpha subscript n equals 1

rightwards double arrow fraction numerator cos alpha subscript 1 over denominator sin alpha subscript 1 end fraction space. fraction numerator cos alpha subscript 2 over denominator sin alpha subscript 2 end fraction space. fraction numerator cos alpha subscript 3 over denominator sin alpha subscript 3 end fraction..... fraction numerator cos alpha subscript n over denominator sin alpha subscript n end fraction equals 1 rightwards double arrow cos alpha subscript 1. space cos alpha subscript 2. space cos alpha subscript 3...... cos alpha subscript n equals sin alpha subscript 1. space sin alpha subscript 2. space sin alpha subscript 3...... sin alpha subscript n t h i s space i s space p o s s i b l e space o n l y space w h e n space alpha subscript 1 equals alpha subscript 2 equals space alpha subscript 3 equals..... alpha subscript n equals 45 degree S o comma space cos alpha subscript 1. space cos alpha subscript 2. space cos alpha subscript 3...... cos alpha subscript n equals fraction numerator 1 over denominator square root of 2 end fraction. fraction numerator 1 over denominator square root of 2 end fraction. fraction numerator 1 over denominator square root of 2 end fraction. fraction numerator 1 over denominator square root of 2 end fraction....... fraction numerator 1 over denominator square root of 2 end fraction equals 1 over 2 to the power of begin display style n over 2 end style end exponent

parallel

Let $P equals left curly bracket theta colon sin space theta minus cos space theta equals square root of 2 cos space theta right curly bracket text and end text Q equals left curly bracket theta colon sin space theta plus cos space theta equals square root of 2 sin space theta right curly bracket$ be two sets. Then

P equals left curly bracket theta colon sin space theta minus cos space theta equals square root of 2 cos space theta right curly bracket text  and  end text Q equals left curly bracket theta colon sin space theta plus cos space theta equals square root of 2 sin space theta right curly bracket

F r o m space P comma space sin theta space minus cos theta equals square root of 2 cos theta space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space F r o m space Q comma space sin theta plus cos theta equals square root of 2 sin theta rightwards double arrow sin theta equals square root of 2 cos theta plus cos theta space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow cos theta equals square root of 2 sin theta minus sin theta rightwards double arrow sin theta equals cos theta open parentheses square root of 2 plus 1 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow cos theta equals sin theta open parentheses square root of 2 minus 1 close parentheses rightwards double arrow fraction numerator sin theta over denominator cos theta end fraction equals square root of 2 plus 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow fraction numerator sin theta over denominator cos theta end fraction equals fraction numerator 1 over denominator square root of 2 minus 1 end fraction cross times fraction numerator square root of 2 plus 1 over denominator square root of 2 plus 1 end fraction rightwards double arrow tan theta equals square root of 2 plus 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow tan theta equals fraction numerator square root of 2 plus 1 over denominator 2 minus 1 end fraction equals square root of 2 plus 1 H e n c e comma space P equals Q

Let $P equals left curly bracket theta colon sin space theta minus cos space theta equals square root of 2 cos space theta right curly bracket text and end text Q equals left curly bracket theta colon sin space theta plus cos space theta equals square root of 2 sin space theta right curly bracket$ be two sets. Then

P equals left curly bracket theta colon sin space theta minus cos space theta equals square root of 2 cos space theta right curly bracket text  and  end text Q equals left curly bracket theta colon sin space theta plus cos space theta equals square root of 2 sin space theta right curly bracket

F r o m space P comma space sin theta space minus cos theta equals square root of 2 cos theta space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space F r o m space Q comma space sin theta plus cos theta equals square root of 2 sin theta rightwards double arrow sin theta equals square root of 2 cos theta plus cos theta space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow cos theta equals square root of 2 sin theta minus sin theta rightwards double arrow sin theta equals cos theta open parentheses square root of 2 plus 1 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow cos theta equals sin theta open parentheses square root of 2 minus 1 close parentheses rightwards double arrow fraction numerator sin theta over denominator cos theta end fraction equals square root of 2 plus 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow fraction numerator sin theta over denominator cos theta end fraction equals fraction numerator 1 over denominator square root of 2 minus 1 end fraction cross times fraction numerator square root of 2 plus 1 over denominator square root of 2 plus 1 end fraction rightwards double arrow tan theta equals square root of 2 plus 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow tan theta equals fraction numerator square root of 2 plus 1 over denominator 2 minus 1 end fraction equals square root of 2 plus 1 H e n c e comma space P equals Q

If $0 less than alpha less than pi over 6$ then $alpha left parenthesis cosec space alpha right parenthesis$ is

g i v e n comma space 0 less than alpha less than straight pi over 6..... space e q u a t i o n space 1 rightwards double arrow sin 0 less than sin alpha less than sin straight pi over 6 rightwards double arrow 0 less than sin alpha less than 1 half....... space e q u a t i o n space 2 d i v i d i n g space e q u a t i o n space 1 space a n d space 2 rightwards double arrow 0 less than fraction numerator alpha over denominator sin alpha end fraction less than straight pi over 3 rightwards double arrow 0 less than alpha. cos e c alpha less than straight pi over 3 S o comma space alpha space cos e c alpha space less than straight pi over 3 space

If $0 less than alpha less than pi over 6$ then $alpha left parenthesis cosec space alpha right parenthesis$ is

g i v e n comma space 0 less than alpha less than straight pi over 6..... space e q u a t i o n space 1 rightwards double arrow sin 0 less than sin alpha less than sin straight pi over 6 rightwards double arrow 0 less than sin alpha less than 1 half....... space e q u a t i o n space 2 d i v i d i n g space e q u a t i o n space 1 space a n d space 2 rightwards double arrow 0 less than fraction numerator alpha over denominator sin alpha end fraction less than straight pi over 3 rightwards double arrow 0 less than alpha. cos e c alpha less than straight pi over 3 S o comma space alpha space cos e c alpha space less than straight pi over 3 space

A long cylindrical tube carries a highly polished piston and has a side opening. A tuning fork of frequency n is sounded at the sound heard by the listener charges if the piston is moves in or out. At a particular position of the piston is moved through a distance of 9 cm, the intensity of sound becomes minimum, if the speed of sound is 360 m/s, the value of n is

When piston moves a distance

x subscript 1 end subscript

, path difference change by 2 xs.

therefore

the path difference between maxima and consecutive minima=

lambda divided by 2

2 x equals lambda divided by 2

Or
λ=4x=4×9 cm=36cm=0.36m

n equals fraction numerator v over denominator lambda end fraction equals fraction numerator 360 over denominator 0.36 end fraction equals 1000 blank H z

A long cylindrical tube carries a highly polished piston and has a side opening. A tuning fork of frequency n is sounded at the sound heard by the listener charges if the piston is moves in or out. At a particular position of the piston is moved through a distance of 9 cm, the intensity of sound becomes minimum, if the speed of sound is 360 m/s, the value of n is

physics-General

When piston moves a distance

x subscript 1 end subscript

, path difference change by 2 xs.

therefore

the path difference between maxima and consecutive minima=

lambda divided by 2

2 x equals lambda divided by 2

Or
λ=4x=4×9 cm=36cm=0.36m

n equals fraction numerator v over denominator lambda end fraction equals fraction numerator 360 over denominator 0.36 end fraction equals 1000 blank H z

parallel

The value of $cot space fraction numerator 7 pi over denominator 16 end fraction plus 2 cot space fraction numerator 3 pi over denominator 8 end fraction plus cot space fraction numerator 15 pi over denominator 16 end fraction$