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

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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 .

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

B .

\frac{a^{3} b^{3}}{(a + b)^{3}}

C .

\frac{a^{2} b^{2}}{(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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