If $\sin x + \sin^2 x + \sin^3 x = 1$, then $\cos^6 x - 4\cos^4 x + 8\cos^2 x$ = _______.
Step-by-Step Solution
Key Concept: Factoring the original equation strategically and using Pythagorean identities transforms a cubic trigonometric equation into a polynomial in $\cos^2 x$.
From $\sin x + \sin^2 x + \sin^3 x = 1$, factor as $\sin x(1 + \sin^2 x) = 1 - \sin^2 x = \cos^2 x$. Squaring both sides gives $\sin^2 x(1 + \sin^2 x)^2 = \cos^4 x$. After expanding and using $\sin^2 x + \cos^2 x = 1$, this reduces to $\cos^6 x - 4\cos^4 x + 8\cos^2 x = 4$.
Correct Answer: 4