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Question: If \(\sin x + \sin^{2}x = 1,\) then \(\cos^{8}x + 2\cos^{6}x + \cos^{4}x =\)...

If sin⁡x+sin⁡2x=1,\sin x + \sin^{2}x = 1, then cos⁡8x+2cos⁡6x+cos⁡4x=\cos^{8}x + 2\cos^{6}x + \cos^{4}x =

A

0

B

– 1

C

2

D

1

Answer

1

Explanation

Solution

We have sin⁡x+sin⁡2x=1⇒sin⁡x=cos⁡2x\sin x + \sin^{2}x = 1 \Rightarrow \sin x = \cos^{2}x

∴ cos⁡8x+2cos⁡6x+cos⁡4x=sin⁡4x+2sin⁡3x+sin⁡2x\cos^{8}x + 2\cos^{6}x + \cos^{4}x = \sin^{4}x + 2\sin^{3}x + \sin^{2}x

=(sin⁡x+sin⁡2x)2=1= (\sin x + \sin^{2}x)^{2} = 1.