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Maths-

General

Easy

Question

If $fraction numerator sin to the power of 4 space theta over denominator a end fraction plus fraction numerator cos to the power of 4 begin display style space end style theta over denominator b end fraction equals fraction numerator 1 over denominator a plus b end fraction text then end text fraction numerator sin to the power of 8 begin display style text end text end style theta over denominator a cubed end fraction plus fraction numerator cos to the power of 8 begin display style space end style theta over denominator b cubed end fraction equals$

$\frac{1}{(a + b)^{3}}$
$\frac{1}{(a + b)^{2}}$
$a + b$
$\frac{1}{a + b}$

hint Hint:

In this question, we have to find the value of $fraction numerator sin to the power of 8 begin display style text end text end style theta over denominator a cubed end fraction plus fraction numerator cos to the power of 8 begin display style space end style theta over denominator b cubed end fraction$ , if it is given that $fraction numerator sin to the power of 4 space theta over denominator a end fraction plus fraction numerator cos to the power of 4 begin display style space end style theta over denominator b end fraction equals fraction numerator 1 over denominator a plus b end fraction$ . First we will find the value of any trigonometric function using the given equation, for this question we will find the value of either sine or cosine function. Later we will substitute the value of the derived function to find the other required function and later solve it to get the value of required function.

The correct answer is: $fraction numerator 1 over denominator left parenthesis a plus b right parenthesis cubed end fraction$

$fraction numerator sin to the power of 4 space theta over denominator a end fraction plus fraction numerator cos to the power of 4 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 b sin to the power of 4 theta plus a open parentheses 1 minus sin squared theta close parentheses squared over denominator a b end fraction equals fraction numerator 1 over denominator a plus b end fraction space rightwards double arrow fraction numerator b sin to the power of 4 theta plus a plus a sin to the power of 4 theta minus 2 a sin squared theta over denominator a b end fraction equals fraction numerator 1 over denominator a plus b end fraction rightwards double arrow open parentheses a plus b close parentheses sin to the power of 4 theta minus 2 a space sin squared theta equals fraction numerator a b over denominator a plus b end fraction minus a rightwards double arrow open parentheses a plus b close parentheses sin to the power of 4 theta minus 2 a space sin squared theta equals fraction numerator a b minus a squared minus a b over denominator a plus b end fraction rightwards double arrow open parentheses a plus b close parentheses sin to the power of 4 theta minus 2 a space sin squared theta equals fraction numerator negative a squared over denominator a plus b end fraction rightwards double arrow open parentheses a plus b close parentheses squared sin to the power of 4 theta minus 2 a left parenthesis a plus b right parenthesis space sin squared theta plus a squared equals 0 rightwards double arrow open parentheses left parenthesis a plus b right parenthesis sin squared theta minus a close parentheses squared equals 0 rightwards double arrow left parenthesis a plus b right parenthesis sin squared theta minus a equals 0 rightwards double arrow sin squared theta equals fraction numerator a over denominator a plus b end fraction N o w comma space fraction numerator sin to the power of 8 space theta over denominator a cubed end fraction plus fraction numerator cos to the power of 8 begin display style space end style theta over denominator b cubed end fraction equals fraction numerator begin display style a to the power of 4 over open parentheses a plus b close parentheses to the power of 4 end style over denominator a cubed end fraction plus fraction numerator open parentheses 1 minus a to the power of 4 over open parentheses a plus b close parentheses to the power of 4 close parentheses over denominator b cubed end fraction equals fraction numerator begin display style a to the power of 4 over open parentheses a plus b close parentheses to the power of 4 end style over denominator a cubed end fraction plus fraction numerator begin display style b to the power of 4 over open parentheses a plus b close parentheses to the power of 4 end style over denominator b cubed end fraction equals a over left parenthesis a plus b right parenthesis to the power of 4 plus b over left parenthesis a plus b right parenthesis to the power of 4 equals fraction numerator a plus b over denominator left parenthesis a plus b right parenthesis to the power of 4 end fraction equals 1 over left parenthesis a plus b right parenthesis cubed$

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