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If $sinA + \sin^{2} A + \sin^{3} A = 1$ , then find the value of $\cos^{6} A - 4 \cos^{4} A + 8 \cos^{2} A$ .

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1

2

3

4

answer is D.

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Detailed Solution

It is given,

sinA + \sin^{2} A + \sin^{3} A = 1

sinA + \sin^{2} A + \sin^{3} A = 1

\Rightarrow sinA (1 + \sin^{2} A) = 1 - \sin^{2} A

\Rightarrow sinA (2 - \cos^{2} A) = \cos^{2} A

Squaring on both sides, we get

\Rightarrow \sin^{2} A (4 - 4 \cos^{2} A + \cos^{4} A) = \cos^{4} A

\Rightarrow (1 - \cos^{2} A) (4 - 4 \cos^{2} A + \cos^{4} A) = \cos^{4} A

\Rightarrow 4 - 8 \cos^{2} A + 5 \cos^{4} A - \cos^{6} A = \cos^{4} A

∴ \cos^{6} A - 4 \cos^{4} A + 8 \cos^{2} A = 4

.
Hence, the correct option is (4).

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