To rewrite $\sin^2(4x) \cos^2(x)$ in terms of the first power of cosine, we can use the power-reducing formulas. The power-reducing formulas are given by:

$\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$ $\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$

Let's start by applying these formulas to each part of the expression.

1. Rewrite $\sin^2(4x)$ using the power-reducing formula for sine: $\sin^2(4x) = \frac{1 - \cos(8x)}{2}$

2. Rewrite $\cos^2(x)$ using the power-reducing formula for cosine: $\cos^2(x) = \frac{1 + \cos(2x)}{2}$

Now, substitute these into the original expression:

$\sin^2(4x) \cos^2(x) = \left( \frac{1 - \cos(8x)}{2} \right) \left( \frac{1 + \cos(2x)}{2} \right)$

Simplify the expression:

$\sin^2(4x) \cos^2(x) = \frac{(1 - \cos(8x))(1 + \cos(2x))}{4}$

Next, use the distributive property to expand the numerator:

$(1 - \cos(8x))(1 + \cos(2x)) = 1 + \cos(2x) - \cos(8x) - \cos(8x) \cos(2x)$

Thus, the expression becomes:

$\sin^2(4x) \cos^2(x) = \frac{1 + \cos(2x) - \cos(8x) - \cos(8x) \cos(2x)}{4}$

We can use a product-to-sum identity to simplify the term $\cos(8x) \cos(2x)$:

$\cos(8x) \cos(2x) = \frac{1}{2} [\cos(10x) + \cos(6x)]$

So the expression is:

$\sin^2(4x) \cos^2(x) = \frac{1 + \cos(2x) - \cos(8x) - \frac{1}{2} (\cos(10x) + \cos(6x))}{4}$

Combine the terms:

$\sin^2(4x) \cos^2(x) = \frac{1 + \cos(2x) - \cos(8x) - \frac{1}{2}\cos(10x) - \frac{1}{2}\cos(6x)}{4}$

Therefore, the simplified form of $\sin^2(4x) \cos^2(x)$ in terms of the first power of cosine is:

$\sin^2(4x) \cos^2(x) = \frac{1}{4} \left( 1 + \cos(2x) - \cos(8x) - \frac{1}{2}\cos(10x) - \frac{1}{2}\cos(6x) \right)$

Would you like more details on any part of this solution or have any further questions?

Here are some related questions you might find useful:

1. How do you derive the power-reducing formulas?
2. What is the product-to-sum identity used for cosine terms?
3. How do you simplify trigonometric expressions involving higher powers of sine and cosine?
4. What are the other common trigonometric identities and how are they used?
5. How do you verify trigonometric identities?
6. What are some real-world applications of trigonometric identities?
7. How can you apply these identities in solving integrals?
8. What is the importance of expressing trigonometric functions in terms of first powers?

Tip: Always start by identifying which trigonometric identities or formulas are applicable to the given problem to simplify your work.