If sin4Aa+cos4Ab=1a+b, then the value of sin8Aa3+cos8Ab3 is equal to

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If

\frac{s i n^{4} A}{a} + \frac{c o s^{4} A}{b} = \frac{1}{a + b},

then the value of

\frac{s i n^{8} A}{a^{3}} + \frac{c o s^{8} A}{b^{3}}

is equal to

Options:

A . 1(a+b)3

B . a3b3(a+b)3

C . a2b2(a+b)2

D . None of these

Answer: Option A
:
A
(a) It is given that

\frac{s i n^{4} A}{a} + \frac{c o s^{4} A}{b} = \frac{1}{a + b}

\Rightarrow \frac{(1 - c o s 2 A)^{2}}{4 a} + \frac{(1 + c o s 2 A)^{2}}{4 b} = \frac{1}{a + b}

\Rightarrow b (a + b) (1 - 2 c o s 2 A + c o s^{2} 2 A) + a (a + b) (1 + 2 c o s 2 A + c o s^{2} 2 A) = 4 a b

\Rightarrow {b (a + b) + a (a + b)} c o s^{2} 2 A + 2 (a + b) (a - b) c o s 2 A

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