To rewrite the expression $\cos^4(x) \sin^2(x)$ in terms of the first power of cosine, we can use the power-reducing identities:

1. For $\cos^2(x)$: $\cos^2(x) = \frac{1 + \cos(2x)}{2}$
2. For $\sin^2(x)$: $\sin^2(x) = \frac{1 - \cos(2x)}{2}$

Applying these identities, we have: $\cos^4(x) = \left(\cos^2(x)\right)^2 = \left(\frac{1 + \cos(2x)}{2}\right)^2 = \frac{(1 + \cos(2x))^2}{4}$ $\sin^2(x) = \frac{1 - \cos(2x)}{2}$

Now, we can rewrite the expression: $\cos^4(x) \sin^2(x) = \frac{(1 + \cos(2x))^2}{4} \cdot \frac{1 - \cos(2x)}{2}$ $= \frac{(1 + \cos(2x))^2 (1 - \cos(2x))}{8}$

Now expand and simplify:

1. Would you like a detailed step-by-step expansion?
2. Do you want to learn how the identities are derived?
3. Are you interested in examples with other trigonometric functions?
4. Should we explore this approach with integrals?
5. Do you need help with similar power-reduction problems?

Tip: Familiarizing yourself with trigonometric identities can simplify many problems in calculus and trigonometry.