The given expression is:

$(\cos(2x))^2 = \text{blank} + \cos(x)$

This problem involves a trigonometric identity. We can use the double angle identity for cosine:

$\cos(2x) = 2\cos^2(x) - 1$

To simplify $(\cos(2x))^2$, let's apply this identity and then expand.

$(\cos(2x))^2 = (2\cos^2(x) - 1)^2$

Now, expanding:

$(2\cos^2(x) - 1)^2 = 4\cos^4(x) - 4\cos^2(x) + 1$

Thus, the equation can be rewritten as:

$(\cos(2x))^2 = 4\cos^4(x) - 4\cos^2(x) + 1$

The blanks in the equation could potentially represent the terms after simplifying it, as needed.

Do you need further breakdown or help with these steps?

Additional Questions:

1. What is the double angle identity for sine?
2. How can we simplify powers of trigonometric functions further?
3. Can you apply these transformations to other trigonometric expressions?
4. What is the Pythagorean identity and how does it relate to this problem?
5. How do higher-order trigonometric identities arise in calculus?

Tip: When dealing with squares of trigonometric functions like $\cos(2x)^2$, look for known identities or expand the terms based on basic algebraic principles.