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If $cosx$ $+ \cos^{2} x = 1$ then $\sin^{12} x + 3 \sin^{10} x +$ $3 \sin^{8} x + \sin^{6} x + \sin^{4} x + 2 \sin^{2} x - 2$ is equal to ?

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0

sinx

1 - cosx

1

answer is C.

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Detailed Solution

Given that,

\Rightarrow cosx = 1 - \cos^{2} x

\Rightarrow cosx = \sin^{2} x

We need to find the value of

\sin^{12} x + 3 \sin^{10} x +

3 \sin^{8} x + \sin^{6} x + \sin^{4} x + 2 \sin^{2} x - 2

.
Now, in the given equation let us substitute

\sin^{2} x = cosx

= cos 6 x + 3 cos 5 x + 3 cos 4 x + cos 3 x + 2 cos 2 x + cos x − 2 = cos 3 x (cos x + 1) 3 + 2 cos 2 x + cos x − 2 = (cos 2 x + cos x) 3 + cos x − 2 (1 − cos 2 x) = (1) 3 + c o s x − 2 c o s x = 1 − c o s x

So, the value of the expression

\sin^{12} x + 3 \sin^{10} x +

3 \sin^{8} x + \sin^{6} x + \sin^{4} x + 2 \sin^{2} x - 2

1 - cosx

.
Hence, the correct option is (3).

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