Question

If $\sin x + \sin^{2}x = 1$, then the value of

$\cos^{12}x + 3\cos^{10}x + 3\cos^{8}x + \cos^{6}x - 2$ is equal to

• A. 0
• B. 1
• C. – 1
• D. 2

Answer 1

Answer: (C)

– 1

Answer 2

We have, $\sin x + \sin^{2}x = 1$

or $\sin x = 1 - \sin^{2}x$ or $\sin x = \cos^{2}x$

∴ $\cos^{12}x + 3\cos^{10}x + 3\cos^{8}x + \cos^{6}x - 2$

$= \sin^{6}x + 3\sin^{5}x + 3\sin^{4}x + \sin^{3}x - 2$

$= (\sin^{2}x)^{3} + 3(\sin^{2}x)^{2}\sin x + 3(\sin^{2}x)(\sin x)^{2} + (\sin x)^{3} - 2$

$= (\sin^{2}x + \sin x)^{3} - 2 = (1)^{3} - 2$ $\lbrack\because\sin x + \sin^{2}x = 1(\text{given})\rbrack$

= – 1.