To prove the identity \( \cos(2x) = \cos^2(x) \), we will start by using the double-angle identity for cosine:

\[
\cos(2x) = 2\cos^2(x) - 1
\]

Now, let's compare this with the identity we need to prove:

\[
\cos(2x) = \cos^2(x)
\]

To check the equality, we'll equate the right-hand side of the double-angle identity to \( \cos^2(x) \):

\[
2\cos^2(x) - 1 = \cos^2(x)
\]

Subtracting \( \cos^2(x) \) from both sides:

\[
\cos^2(x) - 1 = 0
\]

This simplifies to:

\[
\cos^2(x) = 1
\]

Taking the square root of both sides, we get:

\[
\cos(x) = \pm 1
\]

This implies that the identity \( \cos(2x) = \cos^2(x) \) is only true when \( \cos(x) = 1 \) or \( \cos(x) = -1 \).

### Conclusion:
The identity \( \cos(2x) = \cos^2(x) \) is **not true for all values of \( x \)**. It is only true under specific conditions where \( \cos(x) = \pm 1 \).

Would you like to explore the details further or have any questions?

### Related Questions:
1. What are the general forms of trigonometric double-angle identities?
2. How can you derive other forms of the double-angle identity for cosine?
3. What conditions must be met for trigonometric identities to be valid?
4. How does \( \cos(2x) = \cos^2(x) - \sin^2(x) \) relate to other trigonometric identities?
5. What is the graphical interpretation of \( \cos(2x) = \cos^2(x) \) for specific values of \( x \)?

### Tip:
When proving trigonometric identities, always consider the domain and range of the functions involved. This can help identify special cases where an identity might hold true.

Explore the proof of the trigonometric identity cos(2x) = cos²(x) with a detailed step-by-step solution. This problem involves applying the double-angle identity for cosine and understanding the conditions under which the identity holds true. Suitable for advanced high school students and beyond.

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