Biology
General
Easy

Question

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 =

  1. sec squared space theta
  2. c o sec squared space theta
  3. cot squared space theta
  4. tan squared space theta

The correct answer is: c o sec squared space theta

Related Questions to study

General
Maths-

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

Maths-General
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
General
Maths-

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 =

Maths-General
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
General
Maths-

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

Maths-General
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
parallel
General
Maths-

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

Maths-General
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
General
Maths-

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

Given, 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

Maths-General
Given, 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
General
Maths-

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

Given two sets 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

Maths-General
Given two sets 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
parallel
General
Maths-

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

Maths-General
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
General
physics-

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 distancex subscript 1 end subscript, path difference change by 2 xs.
thereforethe 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 distancex subscript 1 end subscript, path difference change by 2 xs.
thereforethe 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
General
Maths-

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

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
equals tan open parentheses straight pi over 2 minus fraction numerator 7 straight pi over denominator 16 end fraction close parentheses plus 2 tan open parentheses straight pi over 2 minus fraction numerator 3 straight pi over denominator 2 end fraction close parentheses plus c o t open parentheses straight pi minus straight pi over 16 close parentheses space space space space open square brackets tan straight pi over 2 minus x equals c o t space x comma space c o t space straight pi minus straight x equals negative cotx close square brackets
equals tan straight pi over 16 plus 2 tan straight pi over 8 minus c o t straight pi over 16
equals open parentheses tan straight pi over 16 minus c o t straight pi over 16 close parentheses plus 2 tan straight pi over 8
equals open parentheses fraction numerator sin begin display style straight pi over 16 end style over denominator cos begin display style straight pi over 16 end style end fraction minus fraction numerator cos begin display style straight pi over 16 end style over denominator sin begin display style straight pi over 16 end style end fraction close parentheses plus 2 fraction numerator sin begin display style straight pi over 8 end style over denominator cos begin display style straight pi over 8 end style end fraction
equals open parentheses fraction numerator sin squared begin display style straight pi over 16 end style minus cos squared begin display style straight pi over 16 end style over denominator cos begin display style straight pi over 16 end style sin begin display style straight pi over 16 end style end fraction close parentheses plus 2 fraction numerator sin begin display style straight pi over 8 end style over denominator cos begin display style straight pi over 8 end style end fraction
equals 2 fraction numerator negative cos begin display style straight pi over 8 end style over denominator sin begin display style straight pi over 8 end style end fraction plus 2 fraction numerator sin begin display style straight pi over 8 end style over denominator cos begin display style straight pi over 8 end style 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 open square brackets cos squared x minus sin squared x equals c o x 2 x comma space 2 sin x space cos x space equals sin 2 x close square brackets
equals negative 2 open parentheses fraction numerator cos squared begin display style straight pi over 8 end style minus sin squared begin display style straight pi over 8 end style over denominator sin begin display style straight pi over 8 end style cos begin display style straight pi over 8 end style end fraction close parentheses
equals negative 2 open parentheses fraction numerator 2 cos begin display style straight pi over 4 end style over denominator sin begin display style straight pi over 4 end style end fraction close parentheses space space space space space space space space space space space space space space space space space space space open square brackets cos squared x minus sin squared x equals c o x 2 x comma space 2 sin x space cos x space equals sin 2 x close square brackets
equals negative 4 c o t straight pi over 4
equals negative 4 cross times 1 space space space space space space space space space space space space space space open square brackets c o t straight pi over 4 equals 1 close square brackets
equals negative 4

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

Maths-General
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
equals tan open parentheses straight pi over 2 minus fraction numerator 7 straight pi over denominator 16 end fraction close parentheses plus 2 tan open parentheses straight pi over 2 minus fraction numerator 3 straight pi over denominator 2 end fraction close parentheses plus c o t open parentheses straight pi minus straight pi over 16 close parentheses space space space space open square brackets tan straight pi over 2 minus x equals c o t space x comma space c o t space straight pi minus straight x equals negative cotx close square brackets
equals tan straight pi over 16 plus 2 tan straight pi over 8 minus c o t straight pi over 16
equals open parentheses tan straight pi over 16 minus c o t straight pi over 16 close parentheses plus 2 tan straight pi over 8
equals open parentheses fraction numerator sin begin display style straight pi over 16 end style over denominator cos begin display style straight pi over 16 end style end fraction minus fraction numerator cos begin display style straight pi over 16 end style over denominator sin begin display style straight pi over 16 end style end fraction close parentheses plus 2 fraction numerator sin begin display style straight pi over 8 end style over denominator cos begin display style straight pi over 8 end style end fraction
equals open parentheses fraction numerator sin squared begin display style straight pi over 16 end style minus cos squared begin display style straight pi over 16 end style over denominator cos begin display style straight pi over 16 end style sin begin display style straight pi over 16 end style end fraction close parentheses plus 2 fraction numerator sin begin display style straight pi over 8 end style over denominator cos begin display style straight pi over 8 end style end fraction
equals 2 fraction numerator negative cos begin display style straight pi over 8 end style over denominator sin begin display style straight pi over 8 end style end fraction plus 2 fraction numerator sin begin display style straight pi over 8 end style over denominator cos begin display style straight pi over 8 end style 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 open square brackets cos squared x minus sin squared x equals c o x 2 x comma space 2 sin x space cos x space equals sin 2 x close square brackets
equals negative 2 open parentheses fraction numerator cos squared begin display style straight pi over 8 end style minus sin squared begin display style straight pi over 8 end style over denominator sin begin display style straight pi over 8 end style cos begin display style straight pi over 8 end style end fraction close parentheses
equals negative 2 open parentheses fraction numerator 2 cos begin display style straight pi over 4 end style over denominator sin begin display style straight pi over 4 end style end fraction close parentheses space space space space space space space space space space space space space space space space space space space open square brackets cos squared x minus sin squared x equals c o x 2 x comma space 2 sin x space cos x space equals sin 2 x close square brackets
equals negative 4 c o t straight pi over 4
equals negative 4 cross times 1 space space space space space space space space space space space space space space open square brackets c o t straight pi over 4 equals 1 close square brackets
equals negative 4
parallel
General
Maths-

For any real theta, the maximum value of cos squared space left parenthesis cos space theta right parenthesis plus sin squared space left parenthesis sin space theta right parenthesis is

As cos is a decreasing function and sin is an increasing function so the function cos squared space left parenthesis cos space theta right parenthesis plus sin squared space left parenthesis sin space theta right parenthesis is maximum when theta=90 degree.
So, the maximum value of cos squared space left parenthesis cos space theta right parenthesis plus sin squared space left parenthesis sin space theta right parenthesis = cos squared space left parenthesis cos space 90 degree right parenthesis plus sin squared space left parenthesis sin space 90 degree right parenthesis equals cos squared 0 plus sin squared 1 equals 1 plus sin squared 1

For any real theta, the maximum value of cos squared space left parenthesis cos space theta right parenthesis plus sin squared space left parenthesis sin space theta right parenthesis is

Maths-General
As cos is a decreasing function and sin is an increasing function so the function cos squared space left parenthesis cos space theta right parenthesis plus sin squared space left parenthesis sin space theta right parenthesis is maximum when theta=90 degree.
So, the maximum value of cos squared space left parenthesis cos space theta right parenthesis plus sin squared space left parenthesis sin space theta right parenthesis = cos squared space left parenthesis cos space 90 degree right parenthesis plus sin squared space left parenthesis sin space 90 degree right parenthesis equals cos squared 0 plus sin squared 1 equals 1 plus sin squared 1
General
chemistry-

Identify A and B

With H2 / pd / CaCO3 - is addition of ‘H’ know is oxidizing agent

Identify A and B

chemistry-General
With H2 / pd / CaCO3 - is addition of ‘H’ know is oxidizing agent
General
Maths-

If square root of 2 cos space A equals cos space B plus cos cubed space B text  and  end text square root of 2 sin text end text A equals sin space B minus sin cubed space B then sin (A-B) =

GIVEN-
​   √2 cos cos cos3 B..... equation 1
∴      cos = 1/√2 (cos cos3 B) ........ (Dividing both sides by √2)

√2 sin sin B - sin3 B..... equation 2

∴      sin = 1/√2 (sin B - sin3 B) ........ (Dividing both sides by √2)
TO FIND-
Sin (A - B)
SOLUTION-
We know that-
Sin (A - B) = (Sin A * Cos B) - (Cos A * Sin B)
∴ Sin (A - B) = [1/√2 (sin B - sin3 B) * cos B] - [1/√2 (cos cos3 B) * sin B] ...... (From Equations i & ii)
∴ Sin (A - B) = [1/√2 * (sin B Cos B - sin3 B Cos B)] - [1/√2 * (sin B cos cos3 B * sin B)]
∴ Sin (A - B) = 1/√2 sin B Cos B - 1/√2 sin3 B Cos B - (1/√2 sin B cos + 1/√2 cos3 B * sin B)
∴ Sin (A - B) = 1/√2 sin B Cos B - 1/√2 sin3 B Cos B - 1/√2 sin B cos - 1/√2 cos3 B * sin B
∴ Sin (A - B) = 1/√2 sin B Cos B - 1/√2 sin3 B Cos B - 1/√2 sin B cos B - 1/√2 cos3 B * sin B
∴ Sin (A - B) = -1/√2 sin3 B Cos B - 1/√2 cos3 B * sin B

∴ Sin (A - B) = -1/√2 sin B Cos B (sin2 B +cos2 B)
                    = -1/2√2 sin2B
Now squaring and adding equation 1 and 2 we get
2 cos squared A plus 2 sin squared A equals cos squared B plus cos to the power of 6 B plus 2 cos to the power of 4 B plus sin squared B plus sin to the power of 6 B minus 2 sin to the power of 4 B
rightwards double arrow 2 open parentheses cos squared A plus sin squared A close parentheses equals cos squared B plus sin squared B plus open parentheses cos squared B plus sin squared B close parentheses cubed minus 3 open parentheses cos squared B sin squared B close parentheses open parentheses cos squared B plus sin squared B close parentheses plus 2 open parentheses cos to the power of 4 B minus sin to the power of 4 B close parentheses
rightwards double arrow 2 equals 1 plus 1 minus 3 over 4 sin squared 2 B plus 2 cos 2 B space space space space open square brackets 4 cos squared B sin squared B equals sin squared 2 B comma cos to the power of 4 B minus sin to the power of 4 B equals cos 2 B space close square brackets space space space open square brackets cos squared B plus sin squared B equals 1 close square brackets
rightwards double arrow 0 equals negative 3 sin squared 2 B plus 8 cos 2 B
rightwards double arrow 0 equals negative 3 open parentheses 1 minus cos squared 2 B close parentheses plus 8 cos 2 B space space space space open square brackets cos squared 2 B plus sin squared 2 B equals 1 close square brackets
rightwards double arrow 0 equals 3 cos squared 2 B minus 3 plus 8 cos 2 B
rightwards double arrow 3 cos squared 2 B plus 8 cos 2 B minus 3 equals 0
rightwards double arrow 3 cos squared 2 B plus 9 cos 2 B minus cos 2 B minus 3 equals 0
rightwards double arrow 3 cos 2 B open parentheses cos 2 B plus 3 close parentheses minus 1 open parentheses cos 2 B plus 3 close parentheses equals 0
rightwards double arrow open parentheses 3 cos 2 B minus 1 close parentheses open parentheses cos 2 B plus 3 close parentheses equals 0
rightwards double arrow cos 2 B equals 1 third space o r space cos 2 B equals negative 3 open parentheses n o t space p o s s i b l e close parentheses
N o w comma space w e space k n o w space t h a t space open parentheses cos squared 2 B plus sin squared 2 B close parentheses equals 1
S o comma space 1 over 9 plus sin squared 2 B equals 1 rightwards double arrow sin squared 2 B equals 1 minus 1 over 9 equals 8 over 9
sin 2 B equals plus-or-minus fraction numerator 2 square root of 2 over denominator 3 end fraction
t h e r e f o r e comma space sin open parentheses A minus B close parentheses equals fraction numerator negative sin 2 B over denominator 2 square root of 2 end fraction equals plus-or-minus fraction numerator 2 square root of 2 over denominator 3 end fraction cross times fraction numerator 1 over denominator 2 square root of 2 end fraction equals plus-or-minus 1 third

If square root of 2 cos space A equals cos space B plus cos cubed space B text  and  end text square root of 2 sin text end text A equals sin space B minus sin cubed space B then sin (A-B) =

Maths-General
GIVEN-
​   √2 cos cos cos3 B..... equation 1
∴      cos = 1/√2 (cos cos3 B) ........ (Dividing both sides by √2)

√2 sin sin B - sin3 B..... equation 2

∴      sin = 1/√2 (sin B - sin3 B) ........ (Dividing both sides by √2)
TO FIND-
Sin (A - B)
SOLUTION-
We know that-
Sin (A - B) = (Sin A * Cos B) - (Cos A * Sin B)
∴ Sin (A - B) = [1/√2 (sin B - sin3 B) * cos B] - [1/√2 (cos cos3 B) * sin B] ...... (From Equations i & ii)
∴ Sin (A - B) = [1/√2 * (sin B Cos B - sin3 B Cos B)] - [1/√2 * (sin B cos cos3 B * sin B)]
∴ Sin (A - B) = 1/√2 sin B Cos B - 1/√2 sin3 B Cos B - (1/√2 sin B cos + 1/√2 cos3 B * sin B)
∴ Sin (A - B) = 1/√2 sin B Cos B - 1/√2 sin3 B Cos B - 1/√2 sin B cos - 1/√2 cos3 B * sin B
∴ Sin (A - B) = 1/√2 sin B Cos B - 1/√2 sin3 B Cos B - 1/√2 sin B cos B - 1/√2 cos3 B * sin B
∴ Sin (A - B) = -1/√2 sin3 B Cos B - 1/√2 cos3 B * sin B

∴ Sin (A - B) = -1/√2 sin B Cos B (sin2 B +cos2 B)
                    = -1/2√2 sin2B
Now squaring and adding equation 1 and 2 we get
2 cos squared A plus 2 sin squared A equals cos squared B plus cos to the power of 6 B plus 2 cos to the power of 4 B plus sin squared B plus sin to the power of 6 B minus 2 sin to the power of 4 B
rightwards double arrow 2 open parentheses cos squared A plus sin squared A close parentheses equals cos squared B plus sin squared B plus open parentheses cos squared B plus sin squared B close parentheses cubed minus 3 open parentheses cos squared B sin squared B close parentheses open parentheses cos squared B plus sin squared B close parentheses plus 2 open parentheses cos to the power of 4 B minus sin to the power of 4 B close parentheses
rightwards double arrow 2 equals 1 plus 1 minus 3 over 4 sin squared 2 B plus 2 cos 2 B space space space space open square brackets 4 cos squared B sin squared B equals sin squared 2 B comma cos to the power of 4 B minus sin to the power of 4 B equals cos 2 B space close square brackets space space space open square brackets cos squared B plus sin squared B equals 1 close square brackets
rightwards double arrow 0 equals negative 3 sin squared 2 B plus 8 cos 2 B
rightwards double arrow 0 equals negative 3 open parentheses 1 minus cos squared 2 B close parentheses plus 8 cos 2 B space space space space open square brackets cos squared 2 B plus sin squared 2 B equals 1 close square brackets
rightwards double arrow 0 equals 3 cos squared 2 B minus 3 plus 8 cos 2 B
rightwards double arrow 3 cos squared 2 B plus 8 cos 2 B minus 3 equals 0
rightwards double arrow 3 cos squared 2 B plus 9 cos 2 B minus cos 2 B minus 3 equals 0
rightwards double arrow 3 cos 2 B open parentheses cos 2 B plus 3 close parentheses minus 1 open parentheses cos 2 B plus 3 close parentheses equals 0
rightwards double arrow open parentheses 3 cos 2 B minus 1 close parentheses open parentheses cos 2 B plus 3 close parentheses equals 0
rightwards double arrow cos 2 B equals 1 third space o r space cos 2 B equals negative 3 open parentheses n o t space p o s s i b l e close parentheses
N o w comma space w e space k n o w space t h a t space open parentheses cos squared 2 B plus sin squared 2 B close parentheses equals 1
S o comma space 1 over 9 plus sin squared 2 B equals 1 rightwards double arrow sin squared 2 B equals 1 minus 1 over 9 equals 8 over 9
sin 2 B equals plus-or-minus fraction numerator 2 square root of 2 over denominator 3 end fraction
t h e r e f o r e comma space sin open parentheses A minus B close parentheses equals fraction numerator negative sin 2 B over denominator 2 square root of 2 end fraction equals plus-or-minus fraction numerator 2 square root of 2 over denominator 3 end fraction cross times fraction numerator 1 over denominator 2 square root of 2 end fraction equals plus-or-minus 1 third
parallel
General
chemistry-

Observer the following reactions and predict the nature of A and B

Conceptual

Observer the following reactions and predict the nature of A and B

chemistry-General
Conceptual
General
chemistry-

on ozonolysis gives

on ozonolysis gives

chemistry-General
General
chemistry-

Which of the following reactions will yield 2, 2-dibromopropane?

Which of the following reactions will yield 2, 2-dibromopropane?

chemistry-General
parallel

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