Q119 / 180 · COMEDK 2025 (10 May Shift 3) · Mathematics
If $\sin x+\sin ^2 x=1$ then $\cos ^8 x+2 \cos ^6 x+\cos ^4 x$ is equal to
- A$0$
- Correct answerB$1$
- C$-1$
- D$2$
Solution
Given $\sin x + \sin^{2} x = 1$.
This implies $\sin x = 1 - \sin^{2} x = \cos^{2} x$.
We need to evaluate $\cos^{8} x + 2 \cos^{6} x + \cos^{4} x$.
Substituting $\cos^{2} x = \sin x$, we get:
$(\cos^{2} x)^{4} + 2(\cos^{2} x)^{3} + (\cos^{2} x)^{2} = (\sin x)^{4} + 2(\sin x)^{3} + (\sin x)^{2}$.
Factoring the expression, we have $(\sin^{2} x + \sin x)^{2}$.
Since $\sin x + \sin^{2} x = 1$, the expression becomes $(1)^{2} = 1$.
Answer: $1$
This implies $\sin x = 1 - \sin^{2} x = \cos^{2} x$.
We need to evaluate $\cos^{8} x + 2 \cos^{6} x + \cos^{4} x$.
Substituting $\cos^{2} x = \sin x$, we get:
$(\cos^{2} x)^{4} + 2(\cos^{2} x)^{3} + (\cos^{2} x)^{2} = (\sin x)^{4} + 2(\sin x)^{3} + (\sin x)^{2}$.
Factoring the expression, we have $(\sin^{2} x + \sin x)^{2}$.
Since $\sin x + \sin^{2} x = 1$, the expression becomes $(1)^{2} = 1$.
Answer: $1$
Hence, the correct answer is option: 'B'.