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Question

If $Tan squared space theta equals square root of 3 plus left parenthesis square root of 3 minus 1 right parenthesis Tan space theta$ and $theta$ lies in $open parentheses negative pi over 2 comma pi over 2 close parentheses$ then $theta$ =

$\{\frac{- π}{3}, \frac{π}{4}\}$
$\{\frac{π}{3}, \frac{- π}{4}\}$
$\{\frac{π}{6}, \frac{- π}{3}\}$
$\{\frac{- π}{12}, \frac{π}{12}\}$

Hint:

in this question, we have to find the value of $ϑ$ . Firstly, we will simplify the equation and as it is given that the value will lie between $open parentheses fraction numerator negative pi over denominator 2 end fraction comma pi over 2 close parentheses$ , so we can find the required solution.

The correct answer is: $open curly brackets pi over 3 comma fraction numerator negative pi over denominator 4 end fraction close curly brackets$

$space space space space t an squared space theta equals square root of 3 plus left parenthesis square root of 3 minus 1 right parenthesis t an space theta rightwards double arrow tan squared theta minus square root of 3 tan space theta minus tan space theta minus square root of 3 equals 0 rightwards double arrow tan space theta left parenthesis tan space theta minus square root of 3 right parenthesis space plus space left parenthesis tan space theta space minus square root of 3 right parenthesis equals 0 rightwards double arrow open parentheses tan space theta plus 1 close parentheses open parentheses tan space theta minus square root of 3 close parentheses equals 0 rightwards double arrow tan space theta equals equals negative 1 space o r space tan space theta space equals square root of 3 rightwards double arrow tan space theta space equals tan space open parentheses fraction numerator negative pi over denominator 4 end fraction close parentheses space o r space tan space theta space equals tan pi over 3 space space space space space space space space space space open square brackets a s space theta space l i e s space i n space open parentheses fraction numerator negative pi over denominator 2 end fraction comma pi over 2 close parentheses close square brackets rightwards double arrow theta equals fraction numerator negative pi over denominator 4 end fraction o r pi over 3 S o comma space open curly brackets fraction numerator negative pi over denominator 4 end fraction comma pi over 3 close curly brackets$

Related Questions to study

If , then $Tan space theta plus Sec space theta equals square root of 3$ he principal value of $open parentheses theta plus pi over 6 close parentheses$ is

Tan space theta plus Sec space theta equals square root of 3 rightwards double arrow fraction numerator s i n theta over denominator cos theta end fraction plus fraction numerator 1 over denominator cos theta end fraction equals square root of 3 rightwards double arrow sin theta plus 1 equals square root of 3 cos theta rightwards double arrow square root of 3 cos theta minus sin theta equals 1 rightwards double arrow fraction numerator square root of 3 over denominator 2 end fraction cos theta minus 1 half sin theta equals 1 half rightwards double arrow cos straight pi over 6 cos theta minus sin straight pi over 6 sin theta equals 1 half rightwards double arrow cos open parentheses straight pi over 6 plus theta close parentheses equals cos straight pi over 3 rightwards double arrow theta plus straight pi over 6 equals straight pi over 3

If , then $Tan space theta plus Sec space theta equals square root of 3$ he principal value of $open parentheses theta plus pi over 6 close parentheses$ is

Tan space theta plus Sec space theta equals square root of 3 rightwards double arrow fraction numerator s i n theta over denominator cos theta end fraction plus fraction numerator 1 over denominator cos theta end fraction equals square root of 3 rightwards double arrow sin theta plus 1 equals square root of 3 cos theta rightwards double arrow square root of 3 cos theta minus sin theta equals 1 rightwards double arrow fraction numerator square root of 3 over denominator 2 end fraction cos theta minus 1 half sin theta equals 1 half rightwards double arrow cos straight pi over 6 cos theta minus sin straight pi over 6 sin theta equals 1 half rightwards double arrow cos open parentheses straight pi over 6 plus theta close parentheses equals cos straight pi over 3 rightwards double arrow theta plus straight pi over 6 equals straight pi over 3

In a $straight triangle A B C comma sin to the power of 4 invisible function application A plus sin to the power of 4 invisible function application B plus sin to the power of 4 invisible function application C equals$

In a $straight triangle A B C comma sin to the power of 4 invisible function application A plus sin to the power of 4 invisible function application B plus sin to the power of 4 invisible function application C equals$

If $4 cos squared space theta equals 3 text then end text theta equals negative midline horizontal ellipsis minus negative$

4 space cos squared theta equals 3 rightwards double arrow cos squared theta equals 3 over 4 rightwards double arrow cos space theta space equals fraction numerator square root of 3 over denominator 2 end fraction rightwards double arrow cos space theta equals cos space pi over 6 space o r space cos space fraction numerator 5 pi over denominator 6 end fraction rightwards double arrow theta equals pi over 6 comma fraction numerator 5 pi over denominator 6 end fraction

If $4 cos squared space theta equals 3 text then end text theta equals negative midline horizontal ellipsis minus negative$

4 space cos squared theta equals 3 rightwards double arrow cos squared theta equals 3 over 4 rightwards double arrow cos space theta space equals fraction numerator square root of 3 over denominator 2 end fraction rightwards double arrow cos space theta equals cos space pi over 6 space o r space cos space fraction numerator 5 pi over denominator 6 end fraction rightwards double arrow theta equals pi over 6 comma fraction numerator 5 pi over denominator 6 end fraction

parallel

The value of $sum from k equals 1 to 100 of sin space left parenthesis k x right parenthesis cos space left parenthesis 101 minus k right parenthesis x$ is equal to

sum from k equals 1 to 100 of   sin space left parenthesis k x right parenthesis cos space left parenthesis 101 minus k right parenthesis x S equals sin space x. cos space 100 x space plus space sin space 2 x. cos space 99 x space plus space sin space 3 x. cos space 98 x space plus..... plus sin space 100 x. cos space x space _ _ _ _ _ _ _ _ e q u a t i o n space 2 W r i t t i n g space i n space t h e space r e v e r s e space o r d e r comma space S equals sin space 100 x. cos space x plus space sin space 99 x. cos space 2 x space plus space sin space 98 x. cos space 3 x space plus...... plus sin space x. cos space 100 x space space space _ _ _ _ _ _ _ _ e q u a t i o n space 1 A d d i n g space t h e space e q u a t i o n space 1 space a n d space 2 comma space w e space g e t colon rightwards double arrow 2 S equals left square bracket left parenthesis sin space x. cos space 100 x space plus space sin space 100 x. cos space x right parenthesis plus left parenthesis sin space 2 x. cos space 99 x space plus space sin space 99 x. cos space 2 x right parenthesis plus............ plus left parenthesis sin space 1 o o x. cos space x space plus space sin space x. cos space 100 x right parenthesis right square bracket rightwards double arrow 2 S equals 100 sin left parenthesis x plus 100 x right parenthesis rightwards double arrow S equals 50 sin left parenthesis 101 x right parenthesis

The value of $sum from k equals 1 to 100 of sin space left parenthesis k x right parenthesis cos space left parenthesis 101 minus k right parenthesis x$ is equal to

sum from k equals 1 to 100 of   sin space left parenthesis k x right parenthesis cos space left parenthesis 101 minus k right parenthesis x S equals sin space x. cos space 100 x space plus space sin space 2 x. cos space 99 x space plus space sin space 3 x. cos space 98 x space plus..... plus sin space 100 x. cos space x space _ _ _ _ _ _ _ _ e q u a t i o n space 2 W r i t t i n g space i n space t h e space r e v e r s e space o r d e r comma space S equals sin space 100 x. cos space x plus space sin space 99 x. cos space 2 x space plus space sin space 98 x. cos space 3 x space plus...... plus sin space x. cos space 100 x space space space _ _ _ _ _ _ _ _ e q u a t i o n space 1 A d d i n g space t h e space e q u a t i o n space 1 space a n d space 2 comma space w e space g e t colon rightwards double arrow 2 S equals left square bracket left parenthesis sin space x. cos space 100 x space plus space sin space 100 x. cos space x right parenthesis plus left parenthesis sin space 2 x. cos space 99 x space plus space sin space 99 x. cos space 2 x right parenthesis plus............ plus left parenthesis sin space 1 o o x. cos space x space plus space sin space x. cos space 100 x right parenthesis right square bracket rightwards double arrow 2 S equals 100 sin left parenthesis x plus 100 x right parenthesis rightwards double arrow S equals 50 sin left parenthesis 101 x right parenthesis

If $alpha comma beta comma gamma text are acute angles and end text cos text end text theta equals sin space beta divided by sin space alpha comma cos space ϕ equals sin space gamma divided by sin invisible function application alpha$ and $cos space left parenthesis theta minus ϕ right parenthesis equals sin space beta sin space gamma$ then the value of $tan squared space alpha minus tan squared space beta minus tan squared space gamma$ is equal to

cos space left parenthesis theta minus ϕ right parenthesis equals sin space beta sin space gamma rightwards double arrow open parentheses cos space theta. cos space ϕ minus sin space theta. sin space ϕ close parentheses equals sin space beta. sin space gamma rightwards double arrow open parentheses cos space theta. cos space ϕ minus sin space beta sin space gamma close parentheses equals sin space theta. sin space ϕ S q u a r i n g space b o t h space s i d e s comma space w e space g e t colon rightwards double arrow open parentheses cos space theta. cos space ϕ minus sin space beta sin space gamma close parentheses squared equals sin squared theta sin squared ϕ rightwards double arrow open parentheses fraction numerator sin beta over denominator sin alpha end fraction fraction numerator sin gamma over denominator sin alpha end fraction minus sin beta. sin gamma close parentheses squared equals open parentheses 1 minus cos squared theta close parentheses open parentheses 1 minus cos squared ϕ close parentheses rightwards double arrow fraction numerator sin squared beta sin squared gamma open parentheses 1 minus sin squared alpha close parentheses over denominator sin squared alpha end fraction equals open parentheses 1 minus fraction numerator sin squared beta over denominator sin squared alpha end fraction close parentheses open parentheses 1 minus fraction numerator sin squared gamma over denominator sin squared alpha end fraction close parentheses rightwards double arrow sin squared beta sin squared gamma open parentheses 1 plus sin to the power of 4 alpha minus 2 sin squared alpha close parentheses equals open parentheses sin squared alpha minus sin squared beta close parentheses open parentheses sin squared alpha minus sin squared gamma close parentheses rightwards double arrow sin squared beta sin squared gamma plus sin squared beta sin squared gamma sin to the power of 4 alpha minus 2 sin squared alpha sin squared beta sin squared gamma equals sin to the power of 4 alpha minus sin squared alpha sin squared beta minus sin squared alpha sin squared gamma plus sin squared beta sin squared gamma rightwards double arrow sin to the power of 4 alpha open parentheses 1 minus sin squared beta sin squared gamma close parentheses minus sin squared gamma open parentheses sin squared beta plus sin squared gamma minus 2 sin squared beta sin squared gamma close parentheses equals 0 rightwards double arrow sin to the power of 4 alpha open parentheses 1 minus sin squared beta sin squared gamma close parentheses equals sin squared gamma open parentheses sin squared beta plus sin squared gamma minus 2 sin squared beta sin squared gamma close parentheses rightwards double arrow sin squared alpha equals fraction numerator sin squared beta plus sin squared gamma minus 2 sin squared beta sin squared gamma over denominator 1 minus sin squared beta sin squared gamma end fraction...... e q u a t i o n space 1 rightwards double arrow cos squared alpha equals 1 minus fraction numerator sin squared beta plus sin squared gamma minus 2 sin squared beta sin squared gamma over denominator 1 minus sin squared beta sin squared gamma end fraction rightwards double arrow cos squared alpha equals fraction numerator 1 minus sin squared beta minus sin squared gamma plus sin squared beta sin squared gamma over denominator 1 minus sin squared beta sin squared gamma end fraction..... e q u a t i o n space 2 f r o m space 1 space a n d space 2 tan squared alpha equals fraction numerator sin squared beta plus sin squared gamma minus 2 sin squared beta sin squared gamma over denominator 1 minus sin squared beta minus sin squared gamma plus s in squared beta sin squared gamma end fraction equals fraction numerator sin squared beta minus sin squared beta sin squared gamma plus sin squared gamma minus sin squared beta sin squared gamma over denominator open parentheses 1 minus sin squared beta close parentheses minus sin squared gamma open parentheses 1 minus sin squared beta close parentheses end fraction equals fraction numerator sin squared beta open parentheses 1 minus sin squared gamma close parentheses plus sin squared gamma open parentheses 1 minus sin squared beta close parentheses over denominator open parentheses 1 minus sin squared beta close parentheses open parentheses 1 minus sin squared gamma close parentheses end fraction equals fraction numerator sin squared beta cos squared gamma plus sin squared gamma cos squared beta over denominator cos squared beta cos squared gamma end fraction N o w comma space tan squared alpha minus tan squared beta minus tan squared gamma equals fraction numerator sin squared beta cos squared gamma plus sin squared gamma cos squared beta over denominator cos squared beta cos squared gamma end fraction minus fraction numerator sin squared beta over denominator cos squared beta end fraction minus fraction numerator sin squared gamma over denominator cos squared gamma end fraction equals fraction numerator sin squared beta cos squared gamma plus sin squared gamma cos squared beta minus sin squared beta cos squared gamma minus sin squared gamma cos squared beta over denominator cos squared beta cos squared gamma end fraction equals 0

If $alpha comma beta comma gamma text are acute angles and end text cos text end text theta equals sin space beta divided by sin space alpha comma cos space ϕ equals sin space gamma divided by sin invisible function application alpha$ and $cos space left parenthesis theta minus ϕ right parenthesis equals sin space beta sin space gamma$ then the value of $tan squared space alpha minus tan squared space beta minus tan squared space gamma$ is equal to

cos space left parenthesis theta minus ϕ right parenthesis equals sin space beta sin space gamma rightwards double arrow open parentheses cos space theta. cos space ϕ minus sin space theta. sin space ϕ close parentheses equals sin space beta. sin space gamma rightwards double arrow open parentheses cos space theta. cos space ϕ minus sin space beta sin space gamma close parentheses equals sin space theta. sin space ϕ S q u a r i n g space b o t h space s i d e s comma space w e space g e t colon rightwards double arrow open parentheses cos space theta. cos space ϕ minus sin space beta sin space gamma close parentheses squared equals sin squared theta sin squared ϕ rightwards double arrow open parentheses fraction numerator sin beta over denominator sin alpha end fraction fraction numerator sin gamma over denominator sin alpha end fraction minus sin beta. sin gamma close parentheses squared equals open parentheses 1 minus cos squared theta close parentheses open parentheses 1 minus cos squared ϕ close parentheses rightwards double arrow fraction numerator sin squared beta sin squared gamma open parentheses 1 minus sin squared alpha close parentheses over denominator sin squared alpha end fraction equals open parentheses 1 minus fraction numerator sin squared beta over denominator sin squared alpha end fraction close parentheses open parentheses 1 minus fraction numerator sin squared gamma over denominator sin squared alpha end fraction close parentheses rightwards double arrow sin squared beta sin squared gamma open parentheses 1 plus sin to the power of 4 alpha minus 2 sin squared alpha close parentheses equals open parentheses sin squared alpha minus sin squared beta close parentheses open parentheses sin squared alpha minus sin squared gamma close parentheses rightwards double arrow sin squared beta sin squared gamma plus sin squared beta sin squared gamma sin to the power of 4 alpha minus 2 sin squared alpha sin squared beta sin squared gamma equals sin to the power of 4 alpha minus sin squared alpha sin squared beta minus sin squared alpha sin squared gamma plus sin squared beta sin squared gamma rightwards double arrow sin to the power of 4 alpha open parentheses 1 minus sin squared beta sin squared gamma close parentheses minus sin squared gamma open parentheses sin squared beta plus sin squared gamma minus 2 sin squared beta sin squared gamma close parentheses equals 0 rightwards double arrow sin to the power of 4 alpha open parentheses 1 minus sin squared beta sin squared gamma close parentheses equals sin squared gamma open parentheses sin squared beta plus sin squared gamma minus 2 sin squared beta sin squared gamma close parentheses rightwards double arrow sin squared alpha equals fraction numerator sin squared beta plus sin squared gamma minus 2 sin squared beta sin squared gamma over denominator 1 minus sin squared beta sin squared gamma end fraction...... e q u a t i o n space 1 rightwards double arrow cos squared alpha equals 1 minus fraction numerator sin squared beta plus sin squared gamma minus 2 sin squared beta sin squared gamma over denominator 1 minus sin squared beta sin squared gamma end fraction rightwards double arrow cos squared alpha equals fraction numerator 1 minus sin squared beta minus sin squared gamma plus sin squared beta sin squared gamma over denominator 1 minus sin squared beta sin squared gamma end fraction..... e q u a t i o n space 2 f r o m space 1 space a n d space 2 tan squared alpha equals fraction numerator sin squared beta plus sin squared gamma minus 2 sin squared beta sin squared gamma over denominator 1 minus sin squared beta minus sin squared gamma plus s in squared beta sin squared gamma end fraction equals fraction numerator sin squared beta minus sin squared beta sin squared gamma plus sin squared gamma minus sin squared beta sin squared gamma over denominator open parentheses 1 minus sin squared beta close parentheses minus sin squared gamma open parentheses 1 minus sin squared beta close parentheses end fraction equals fraction numerator sin squared beta open parentheses 1 minus sin squared gamma close parentheses plus sin squared gamma open parentheses 1 minus sin squared beta close parentheses over denominator open parentheses 1 minus sin squared beta close parentheses open parentheses 1 minus sin squared gamma close parentheses end fraction equals fraction numerator sin squared beta cos squared gamma plus sin squared gamma cos squared beta over denominator cos squared beta cos squared gamma end fraction N o w comma space tan squared alpha minus tan squared beta minus tan squared gamma equals fraction numerator sin squared beta cos squared gamma plus sin squared gamma cos squared beta over denominator cos squared beta cos squared gamma end fraction minus fraction numerator sin squared beta over denominator cos squared beta end fraction minus fraction numerator sin squared gamma over denominator cos squared gamma end fraction equals fraction numerator sin squared beta cos squared gamma plus sin squared gamma cos squared beta minus sin squared beta cos squared gamma minus sin squared gamma cos squared beta over denominator cos squared beta cos squared gamma end fraction equals 0

If $x sin space a plus y sin space 2 a plus z sin space 3 a equals sin space 4 a x sin space b plus y sin space 2 b plus z sin space 3 b equals sin space 4 b comma x sin space c plus y sin space 2 c plus z sin space 3 c equals sin space 4 c$ Then the roots of the equation $t cubed minus open parentheses z over 2 close parentheses t squared minus open parentheses fraction numerator y plus z over denominator 4 end fraction close parentheses t plus open parentheses fraction numerator z minus x over denominator 8 end fraction close parentheses equals 0 comma a comma b comma c not equal to n pi$ are

x sin space a plus y sin space 2 a plus z sin space 3 a equals sin space 4 a x sin space b plus y sin space 2 b plus z sin space 3 b equals sin space 4 b x sin space c plus y sin space 2 c plus z sin space 3 c equals sin space 4 c L e t space a comma space b comma space c space a r e space t h e space r o o t s space o f x sin space theta plus y sin space 2 theta plus z sin space 3 theta equals sin space 4 theta rightwards double arrow x space sin space theta plus y left parenthesis 2 sin space theta space cos space theta right parenthesis plus z left parenthesis 3 sin space theta plus 4 space sin cubed theta space right parenthesis equals open parentheses 4 space sin space theta. cos space theta space cos space 2 theta close parentheses rightwards double arrow sin space theta open parentheses x plus 2 y. cos space theta plus z left parenthesis 3 minus sin squared theta right parenthesis close parentheses equals sin space theta left parenthesis 4 cos space theta left parenthesis 2 cos squared minus 1 right parenthesis right parenthesis rightwards double arrow sin space theta open parentheses x plus 2 y. cos space theta plus z left parenthesis 3 minus sin squared theta right parenthesis minus 8 cos cubed theta plus 4 space cos space theta close parentheses equals 0 rightwards double arrow sin space theta open parentheses negative 8 sin cubed theta plus 4 z space cos squared theta plus left parenthesis 2 y plus 4 right parenthesis cos space theta minus 4 z plus x plus 3 z close parentheses equals 0 rightwards double arrow sin space theta open parentheses cos cubed theta minus z over 2 cos squared theta minus fraction numerator left parenthesis y plus 2 right parenthesis over denominator 4 end fraction cos theta minus fraction numerator left parenthesis z minus x right parenthesis over denominator 8 end fraction close parentheses equals 0 rightwards double arrow e i t h e r comma space sin space theta equals 0 space o r space cos cubed theta minus z over 2 cos squared theta minus fraction numerator left parenthesis y plus 2 right parenthesis over denominator 4 end fraction cos theta minus fraction numerator left parenthesis z minus x right parenthesis over denominator 8 end fraction equals 0 space t h e n comma space cos space a comma space cos space b comma space cos space c space a r e space t h e space s o l u t i o n space o f space t cubed minus z over 2 t squared minus fraction numerator left parenthesis y plus 2 right parenthesis over denominator 4 end fraction t minus fraction numerator left parenthesis z minus x right parenthesis over denominator 8 end fraction comma space left parenthesis a comma space b comma space c not equal to n straight pi right parenthesis

If $x sin space a plus y sin space 2 a plus z sin space 3 a equals sin space 4 a x sin space b plus y sin space 2 b plus z sin space 3 b equals sin space 4 b comma x sin space c plus y sin space 2 c plus z sin space 3 c equals sin space 4 c$ Then the roots of the equation $t cubed minus open parentheses z over 2 close parentheses t squared minus open parentheses fraction numerator y plus z over denominator 4 end fraction close parentheses t plus open parentheses fraction numerator z minus x over denominator 8 end fraction close parentheses equals 0 comma a comma b comma c not equal to n pi$ are

x sin space a plus y sin space 2 a plus z sin space 3 a equals sin space 4 a x sin space b plus y sin space 2 b plus z sin space 3 b equals sin space 4 b x sin space c plus y sin space 2 c plus z sin space 3 c equals sin space 4 c L e t space a comma space b comma space c space a r e space t h e space r o o t s space o f x sin space theta plus y sin space 2 theta plus z sin space 3 theta equals sin space 4 theta rightwards double arrow x space sin space theta plus y left parenthesis 2 sin space theta space cos space theta right parenthesis plus z left parenthesis 3 sin space theta plus 4 space sin cubed theta space right parenthesis equals open parentheses 4 space sin space theta. cos space theta space cos space 2 theta close parentheses rightwards double arrow sin space theta open parentheses x plus 2 y. cos space theta plus z left parenthesis 3 minus sin squared theta right parenthesis close parentheses equals sin space theta left parenthesis 4 cos space theta left parenthesis 2 cos squared minus 1 right parenthesis right parenthesis rightwards double arrow sin space theta open parentheses x plus 2 y. cos space theta plus z left parenthesis 3 minus sin squared theta right parenthesis minus 8 cos cubed theta plus 4 space cos space theta close parentheses equals 0 rightwards double arrow sin space theta open parentheses negative 8 sin cubed theta plus 4 z space cos squared theta plus left parenthesis 2 y plus 4 right parenthesis cos space theta minus 4 z plus x plus 3 z close parentheses equals 0 rightwards double arrow sin space theta open parentheses cos cubed theta minus z over 2 cos squared theta minus fraction numerator left parenthesis y plus 2 right parenthesis over denominator 4 end fraction cos theta minus fraction numerator left parenthesis z minus x right parenthesis over denominator 8 end fraction close parentheses equals 0 rightwards double arrow e i t h e r comma space sin space theta equals 0 space o r space cos cubed theta minus z over 2 cos squared theta minus fraction numerator left parenthesis y plus 2 right parenthesis over denominator 4 end fraction cos theta minus fraction numerator left parenthesis z minus x right parenthesis over denominator 8 end fraction equals 0 space t h e n comma space cos space a comma space cos space b comma space cos space c space a r e space t h e space s o l u t i o n space o f space t cubed minus z over 2 t squared minus fraction numerator left parenthesis y plus 2 right parenthesis over denominator 4 end fraction t minus fraction numerator left parenthesis z minus x right parenthesis over denominator 8 end fraction comma space left parenthesis a comma space b comma space c not equal to n straight pi right parenthesis

parallel

If $equals fraction numerator 2 pi over denominator 7 end fraction$ ,then tan $alpha tan invisible function application 2 alpha plus tan invisible function application 2 alpha tan invisible function application 4 alpha plus tan invisible function application 4 alpha tan invisible function application alpha equals$

tan space alpha. tan space 2 alpha plus tan space 2 alpha. tan space 4 alpha plus tan space 4 alpha. tan space alpha equals fraction numerator sin space alpha over denominator cos space alpha end fraction. fraction numerator sin space 2 alpha over denominator cos space 2 alpha end fraction plus fraction numerator sin space 2 alpha over denominator cos space 2 alpha end fraction. fraction numerator sin space 4 alpha over denominator cos space 4 alpha end fraction plus fraction numerator sin space 4 alpha over denominator cos space 4 alpha end fraction. fraction numerator sin space alpha over denominator cos space alpha end fraction equals fraction numerator sin space alpha. sin space 2 alpha. cos space 4 alpha plus cos space alpha. sin space 2 alpha. sin space 4 alpha plus sin space alpha. cos space 2 alpha. sin space 4 alpha over denominator cos space alpha. cos space 2 alpha. cos space 4 alpha end fraction equals 1 minus open parentheses 1 minus fraction numerator sin space alpha. sin space 2 alpha. cos space 4 alpha plus cos space alpha. sin space 2 alpha. sin space 4 alpha plus sin space alpha. cos space 2 alpha. sin space 4 alpha over denominator cos space alpha. cos space 2 alpha. cos space 4 alpha end fraction close parentheses equals 1 minus open parentheses fraction numerator cos space alpha. cos space 2 alpha. cos space 4 alpha plus sin space alpha. sin space 2 alpha. cos space 4 alpha plus cos space alpha. sin space 2 alpha. sin space 4 alpha plus sin space alpha. cos space 2 alpha. sin space 4 alpha over denominator cos space alpha. cos space 2 alpha. cos space 4 alpha end fraction close parentheses equals 1 minus fraction numerator cos space left parenthesis alpha plus 2 alpha plus 4 alpha right parenthesis over denominator cos space alpha. cos space 2 alpha. cos space 4 alpha end fraction equals 1 minus fraction numerator cos space 7 alpha over denominator begin display style fraction numerator sin space 8 alpha over denominator 8 space sin space alpha end fraction end style end fraction equals 1 minus fraction numerator 8. sin space alpha. cos space 7 alpha over denominator sin space 8 alpha end fraction S u b s t i t u t i n g space t h e space v a l u e space o f space alpha equals fraction numerator 2 pi over denominator 7 end fraction comma space equals 1 minus fraction numerator 8. sin space open parentheses fraction numerator 2 pi over denominator 7 end fraction close parentheses. cos space 2 pi over denominator sin space open parentheses fraction numerator 2 pi over denominator 7 end fraction plus 2 pi close parentheses end fraction equals 1 minus 8 equals negative 7

If $equals fraction numerator 2 pi over denominator 7 end fraction$ ,then tan $alpha tan invisible function application 2 alpha plus tan invisible function application 2 alpha tan invisible function application 4 alpha plus tan invisible function application 4 alpha tan invisible function application alpha equals$

tan space alpha. tan space 2 alpha plus tan space 2 alpha. tan space 4 alpha plus tan space 4 alpha. tan space alpha equals fraction numerator sin space alpha over denominator cos space alpha end fraction. fraction numerator sin space 2 alpha over denominator cos space 2 alpha end fraction plus fraction numerator sin space 2 alpha over denominator cos space 2 alpha end fraction. fraction numerator sin space 4 alpha over denominator cos space 4 alpha end fraction plus fraction numerator sin space 4 alpha over denominator cos space 4 alpha end fraction. fraction numerator sin space alpha over denominator cos space alpha end fraction equals fraction numerator sin space alpha. sin space 2 alpha. cos space 4 alpha plus cos space alpha. sin space 2 alpha. sin space 4 alpha plus sin space alpha. cos space 2 alpha. sin space 4 alpha over denominator cos space alpha. cos space 2 alpha. cos space 4 alpha end fraction equals 1 minus open parentheses 1 minus fraction numerator sin space alpha. sin space 2 alpha. cos space 4 alpha plus cos space alpha. sin space 2 alpha. sin space 4 alpha plus sin space alpha. cos space 2 alpha. sin space 4 alpha over denominator cos space alpha. cos space 2 alpha. cos space 4 alpha end fraction close parentheses equals 1 minus open parentheses fraction numerator cos space alpha. cos space 2 alpha. cos space 4 alpha plus sin space alpha. sin space 2 alpha. cos space 4 alpha plus cos space alpha. sin space 2 alpha. sin space 4 alpha plus sin space alpha. cos space 2 alpha. sin space 4 alpha over denominator cos space alpha. cos space 2 alpha. cos space 4 alpha end fraction close parentheses equals 1 minus fraction numerator cos space left parenthesis alpha plus 2 alpha plus 4 alpha right parenthesis over denominator cos space alpha. cos space 2 alpha. cos space 4 alpha end fraction equals 1 minus fraction numerator cos space 7 alpha over denominator begin display style fraction numerator sin space 8 alpha over denominator 8 space sin space alpha end fraction end style end fraction equals 1 minus fraction numerator 8. sin space alpha. cos space 7 alpha over denominator sin space 8 alpha end fraction S u b s t i t u t i n g space t h e space v a l u e space o f space alpha equals fraction numerator 2 pi over denominator 7 end fraction comma space equals 1 minus fraction numerator 8. sin space open parentheses fraction numerator 2 pi over denominator 7 end fraction close parentheses. cos space 2 pi over denominator sin space open parentheses fraction numerator 2 pi over denominator 7 end fraction plus 2 pi close parentheses end fraction equals 1 minus 8 equals negative 7

If the mapping $f left parenthesis x right parenthesis equals a x plus b comma a less than 0$ maps [-1,1] onto [0,2] then for all values of $theta$ , $A equals cos squared space theta plus sin to the power of 4 space theta$ is such that

A equals cos squared space theta plus sin to the power of 4 space theta equals cos squared theta plus open parentheses 1 minus cos squared theta close parentheses squared equals cos squared theta plus 1 plus cos to the power of 4 theta minus 2 cos squared theta equals cos to the power of 4 theta minus cos squared theta plus 1 l e t space cos squared theta space b e space t. S o comma space A equals t squared minus t plus 1 a s space cos squared theta space equals t space s o comma space 0 less or equal than t less or equal than 1 m i n i m u m space v a l u e space equals fraction numerator negative open parentheses b squared minus 4 a c close parentheses over denominator 4 end fraction equals fraction numerator negative open parentheses open parentheses negative 1 close parentheses squared minus 4.1.1 close parentheses over denominator 4 end fraction equals fraction numerator negative open parentheses 1 minus 4 close parentheses over denominator 4 end fraction equals 3 over 4 m a x i m u m space v a l u e equals 0 minus 0 plus 1 space equals 1 t h e r e f o r e comma space r a n g e open square brackets 3 over 4 comma 1 close square brackets a s space f open parentheses 1 fourth close parentheses equals 3 over 4 space a n d space f open parentheses 0 close parentheses equals 1 s o comma space t h e space r e q u i r e d space r a n g e space i s space f open parentheses 1 fourth close parentheses less or equal than A less or equal than f open parentheses 0 close parentheses

If the mapping $f left parenthesis x right parenthesis equals a x plus b comma a less than 0$ maps [-1,1] onto [0,2] then for all values of $theta$ , $A equals cos squared space theta plus sin to the power of 4 space theta$ is such that

A equals cos squared space theta plus sin to the power of 4 space theta equals cos squared theta plus open parentheses 1 minus cos squared theta close parentheses squared equals cos squared theta plus 1 plus cos to the power of 4 theta minus 2 cos squared theta equals cos to the power of 4 theta minus cos squared theta plus 1 l e t space cos squared theta space b e space t. S o comma space A equals t squared minus t plus 1 a s space cos squared theta space equals t space s o comma space 0 less or equal than t less or equal than 1 m i n i m u m space v a l u e space equals fraction numerator negative open parentheses b squared minus 4 a c close parentheses over denominator 4 end fraction equals fraction numerator negative open parentheses open parentheses negative 1 close parentheses squared minus 4.1.1 close parentheses over denominator 4 end fraction equals fraction numerator negative open parentheses 1 minus 4 close parentheses over denominator 4 end fraction equals 3 over 4 m a x i m u m space v a l u e equals 0 minus 0 plus 1 space equals 1 t h e r e f o r e comma space r a n g e open square brackets 3 over 4 comma 1 close square brackets a s space f open parentheses 1 fourth close parentheses equals 3 over 4 space a n d space f open parentheses 0 close parentheses equals 1 s o comma space t h e space r e q u i r e d space r a n g e space i s space f open parentheses 1 fourth close parentheses less or equal than A less or equal than f open parentheses 0 close parentheses

If $fraction numerator sec to the power of 8 space theta over denominator a end fraction plus fraction numerator tan to the power of 8 begin display style space end style theta over denominator b end fraction equals fraction numerator 1 over denominator a plus b end fraction$ then

fraction numerator sec to the power of 8 space theta over denominator a end fraction plus fraction numerator tan to the power of 8 begin display style space end style theta over denominator b end fraction equals fraction numerator 1 over denominator a plus b end fraction rightwards double arrow fraction numerator begin display style fraction numerator 1 over denominator cos to the power of 8 theta end fraction end style over denominator a end fraction plus fraction numerator begin display style fraction numerator sin to the power of 8 theta over denominator cos to the power of 8 theta end fraction end style over denominator b end fraction equals fraction numerator 1 over denominator a plus b end fraction rightwards double arrow fraction numerator b plus a sin to the power of 8 theta over denominator a b cos to the power of 8 theta end fraction equals fraction numerator 1 over denominator a plus b end fraction rightwards double arrow a b plus a squared sin to the power of 8 theta plus b squared plus a b sin to the power of 8 theta equals a b cos to the power of 8 theta rightwards double arrow a b plus a squared sin to the power of 8 theta plus b squared plus a b sin to the power of 8 theta equals a b open parentheses 1 minus sin squared theta close parentheses to the power of 4 rightwards double arrow a b plus a squared sin to the power of 8 theta plus b squared plus a b sin to the power of 8 theta equals a b open parentheses 1 minus 4 sin squared theta plus 6 sin to the power of 4 theta minus 4 sin to the power of 6 theta plus sin to the power of 8 theta close parentheses space space space space space space space space open square brackets open parentheses a minus b close parentheses to the power of 4 equals a squared minus 4 a b squared plus 6 a squared b squared minus 4 a cubed b plus b to the power of 4 close square brackets rightwards double arrow a b plus a squared sin to the power of 8 theta plus b squared plus a b sin to the power of 8 theta equals a b minus 4 a b sin squared theta plus 6 a b sin to the power of 4 theta minus 4 a b sin to the power of 6 theta plus a b sin to the power of 8 theta rightwards double arrow a squared sin to the power of 8 theta plus b squared equals 6 a b sin to the power of 4 theta minus 4 a b sin to the power of 6 theta minus 4 a b sin squared theta rightwards double arrow open parentheses a sin to the power of 4 theta close parentheses squared plus b squared minus 2 a b sin to the power of 4 theta equals 4 a b sin squared theta open parentheses sin squared theta minus 1 minus sin to the power of 4 theta close parentheses rightwards double arrow open parentheses a sin to the power of 4 theta plus b close parentheses squared equals 4 a b sin squared theta open parentheses negative cos squared theta minus sin to the power of 4 theta close parentheses rightwards double arrow open parentheses a sin to the power of 4 theta plus b close parentheses squared equals negative 4 a b sin squared theta open parentheses cos squared theta plus sin to the power of 4 theta close parentheses rightwards double arrow a comma b space a r e space o f space o p p o s i t e space s i g n space rightwards double arrow a b less or equal than 0

If $fraction numerator sec to the power of 8 space theta over denominator a end fraction plus fraction numerator tan to the power of 8 begin display style space end style theta over denominator b end fraction equals fraction numerator 1 over denominator a plus b end fraction$ then

fraction numerator sec to the power of 8 space theta over denominator a end fraction plus fraction numerator tan to the power of 8 begin display style space end style theta over denominator b end fraction equals fraction numerator 1 over denominator a plus b end fraction rightwards double arrow fraction numerator begin display style fraction numerator 1 over denominator cos to the power of 8 theta end fraction end style over denominator a end fraction plus fraction numerator begin display style fraction numerator sin to the power of 8 theta over denominator cos to the power of 8 theta end fraction end style over denominator b end fraction equals fraction numerator 1 over denominator a plus b end fraction rightwards double arrow fraction numerator b plus a sin to the power of 8 theta over denominator a b cos to the power of 8 theta end fraction equals fraction numerator 1 over denominator a plus b end fraction rightwards double arrow a b plus a squared sin to the power of 8 theta plus b squared plus a b sin to the power of 8 theta equals a b cos to the power of 8 theta rightwards double arrow a b plus a squared sin to the power of 8 theta plus b squared plus a b sin to the power of 8 theta equals a b open parentheses 1 minus sin squared theta close parentheses to the power of 4 rightwards double arrow a b plus a squared sin to the power of 8 theta plus b squared plus a b sin to the power of 8 theta equals a b open parentheses 1 minus 4 sin squared theta plus 6 sin to the power of 4 theta minus 4 sin to the power of 6 theta plus sin to the power of 8 theta close parentheses space space space space space space space space open square brackets open parentheses a minus b close parentheses to the power of 4 equals a squared minus 4 a b squared plus 6 a squared b squared minus 4 a cubed b plus b to the power of 4 close square brackets rightwards double arrow a b plus a squared sin to the power of 8 theta plus b squared plus a b sin to the power of 8 theta equals a b minus 4 a b sin squared theta plus 6 a b sin to the power of 4 theta minus 4 a b sin to the power of 6 theta plus a b sin to the power of 8 theta rightwards double arrow a squared sin to the power of 8 theta plus b squared equals 6 a b sin to the power of 4 theta minus 4 a b sin to the power of 6 theta minus 4 a b sin squared theta rightwards double arrow open parentheses a sin to the power of 4 theta close parentheses squared plus b squared minus 2 a b sin to the power of 4 theta equals 4 a b sin squared theta open parentheses sin squared theta minus 1 minus sin to the power of 4 theta close parentheses rightwards double arrow open parentheses a sin to the power of 4 theta plus b close parentheses squared equals 4 a b sin squared theta open parentheses negative cos squared theta minus sin to the power of 4 theta close parentheses rightwards double arrow open parentheses a sin to the power of 4 theta plus b close parentheses squared equals negative 4 a b sin squared theta open parentheses cos squared theta plus sin to the power of 4 theta close parentheses rightwards double arrow a comma b space a r e space o f space o p p o s i t e space s i g n space rightwards double arrow a b less or equal than 0

parallel

The sum of the series, $sin space theta sec space 3 theta plus sin space 3 theta sec space 3 squared theta plus sin space 3 squared theta sec space 3 cubed theta plus times times$ upto n terms, is

sin space theta sec space 3 theta plus sin space 3 theta sec space 3 squared theta plus sin space 3 squared theta sec space 3 cubed theta plus times times

N o w comma space sin theta. s e c 3 theta equals fraction numerator sin theta over denominator cos 3 theta 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 equals fraction numerator 2 sin theta. cos theta over denominator 2 cos 3 theta. cos theta 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 equals 1 half. fraction numerator sin 2 theta over denominator cos theta cos 3 theta 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 equals 1 half open parentheses fraction numerator sin open parentheses 3 theta minus theta close parentheses over denominator cos theta. cos 3 theta 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 equals 1 half open parentheses fraction numerator sin 3 theta cos theta minus cos 3 theta sin theta over denominator cos theta. cos 3 theta 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 equals 1 half open parentheses fraction numerator sin 3 theta over denominator cos theta end fraction minus fraction numerator sin theta over denominator cos theta 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 equals 1 half open parentheses tan 3 theta minus tan theta close parentheses S i m i l a r l y comma space sin 3 theta. s e c 9 theta equals 1 half open parentheses tan 9 theta minus tan theta close parentheses comma space sin 9 theta. s e c 27 theta equals 1 half open parentheses tan 27 theta minus tan 9 theta close parentheses comma space sin 3 to the power of open parentheses n minus 1 close parentheses end exponent theta. s e c 3 to the power of n theta equals 1 half open parentheses tan 3 to the power of n theta minus tan 3 to the power of open parentheses n minus 1 close parentheses end exponent theta close parentheses S o comma space S n equals 1 half open parentheses tan 3 theta minus tan theta close parentheses plus 1 half open parentheses tan 9 theta minus tan theta close parentheses plus 1 half open parentheses tan 27 theta minus tan 9 theta close parentheses plus..... plus 1 half open parentheses tan 3 to the power of n theta minus tan 3 to the power of open parentheses n minus 1 close parentheses end exponent theta close parentheses space space space space equals 1 half open parentheses tan 3 theta minus tan theta plus tan 9 theta minus tan theta plus tan 27 theta minus tan 9 theta plus...... plus tan 3 to the power of n theta minus tan 3 to the power of open parentheses n minus 1 close parentheses end exponent theta close parentheses space space space space equals 1 half open parentheses tan 3 to the power of n theta minus tan theta close parentheses space space space space space space space space space space space space

The sum of the series, $sin space theta sec space 3 theta plus sin space 3 theta sec space 3 squared theta plus sin space 3 squared theta sec space 3 cubed theta plus times times$ upto n terms, is

sin space theta sec space 3 theta plus sin space 3 theta sec space 3 squared theta plus sin space 3 squared theta sec space 3 cubed theta plus times times

N o w comma space sin theta. s e c 3 theta equals fraction numerator sin theta over denominator cos 3 theta 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 equals fraction numerator 2 sin theta. cos theta over denominator 2 cos 3 theta. cos theta 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 equals 1 half. fraction numerator sin 2 theta over denominator cos theta cos 3 theta 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 equals 1 half open parentheses fraction numerator sin open parentheses 3 theta minus theta close parentheses over denominator cos theta. cos 3 theta 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 equals 1 half open parentheses fraction numerator sin 3 theta cos theta minus cos 3 theta sin theta over denominator cos theta. cos 3 theta 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 equals 1 half open parentheses fraction numerator sin 3 theta over denominator cos theta end fraction minus fraction numerator sin theta over denominator cos theta 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 equals 1 half open parentheses tan 3 theta minus tan theta close parentheses S i m i l a r l y comma space sin 3 theta. s e c 9 theta equals 1 half open parentheses tan 9 theta minus tan theta close parentheses comma space sin 9 theta. s e c 27 theta equals 1 half open parentheses tan 27 theta minus tan 9 theta close parentheses comma space sin 3 to the power of open parentheses n minus 1 close parentheses end exponent theta. s e c 3 to the power of n theta equals 1 half open parentheses tan 3 to the power of n theta minus tan 3 to the power of open parentheses n minus 1 close parentheses end exponent theta close parentheses S o comma space S n equals 1 half open parentheses tan 3 theta minus tan theta close parentheses plus 1 half open parentheses tan 9 theta minus tan theta close parentheses plus 1 half open parentheses tan 27 theta minus tan 9 theta close parentheses plus..... plus 1 half open parentheses tan 3 to the power of n theta minus tan 3 to the power of open parentheses n minus 1 close parentheses end exponent theta close parentheses space space space space equals 1 half open parentheses tan 3 theta minus tan theta plus tan 9 theta minus tan theta plus tan 27 theta minus tan 9 theta plus...... plus tan 3 to the power of n theta minus tan 3 to the power of open parentheses n minus 1 close parentheses end exponent theta close parentheses space space space space equals 1 half open parentheses tan 3 to the power of n theta minus tan theta close parentheses space space space space space space space space space space space space

In a triangle $ABC$

We know that GM

less or equal than

open square brackets sin squared A over 2. sin squared B over 2. sin squared C over 2 close square brackets to the power of 1 third end exponent less or equal than fraction numerator sin squared begin display style A over 2 end style plus sin squared begin display style B over 2 end style plus sin squared begin display style C over 2 end style over denominator 3 end fraction A l s o comma space sin squared A over 2 plus sin squared B over 2 plus sin squared C over 2 greater or equal than 3 over 4 open square brackets sin squared A over 2. sin squared B over 2. sin squared C over 2 close square brackets to the power of 1 third end exponent less or equal than 1 third cross times 3 over 4 rightwards double arrow sin squared A over 2. sin squared B over 2. sin squared C over 2 less or equal than open parentheses 1 fourth close parentheses cubed rightwards double arrow open parentheses sin A over 2. sin B over 2. sin C over 2 close parentheses squared less or equal than 1 over 64 rightwards double arrow sin A over 2. sin B over 2. sin C over 2 less or equal than 1 over 8

In a triangle $ABC$

We know that GM

less or equal than

open square brackets sin squared A over 2. sin squared B over 2. sin squared C over 2 close square brackets to the power of 1 third end exponent less or equal than fraction numerator sin squared begin display style A over 2 end style plus sin squared begin display style B over 2 end style plus sin squared begin display style C over 2 end style over denominator 3 end fraction A l s o comma space sin squared A over 2 plus sin squared B over 2 plus sin squared C over 2 greater or equal than 3 over 4 open square brackets sin squared A over 2. sin squared B over 2. sin squared C over 2 close square brackets to the power of 1 third end exponent less or equal than 1 third cross times 3 over 4 rightwards double arrow sin squared A over 2. sin squared B over 2. sin squared C over 2 less or equal than open parentheses 1 fourth close parentheses cubed rightwards double arrow open parentheses sin A over 2. sin B over 2. sin C over 2 close parentheses squared less or equal than 1 over 64 rightwards double arrow sin A over 2. sin B over 2. sin C over 2 less or equal than 1 over 8

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

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

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