To solve the equation \( \cos(2x) - \cos(x) + 1 = 0 \), we can use the double angle identity for cosine, which states that:

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

Substituting this into the equation gives us:

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

This simplifies to:

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

Factoring out \( \cos(x) \):

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

This leads to two equations:

1. \( \cos(x) = 0 \)
2. \( 2\cos(x) - 1 = 0 \)

### Solving each case:

1. **For \( \cos(x) = 0 \):**
   - The solutions are:
     \[
     x = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z})
     \]

2. **For \( 2\cos(x) - 1 = 0 \):**
   - This simplifies to \( \cos(x) = \frac{1}{2} \).
   - The solutions are:
     \[
     x = \frac{\pi}{3} + 2n\pi \quad \text{and} \quad x = \frac{5\pi}{3} + 2n\pi \quad (n \in \mathbb{Z})
     \]

### Summary of Solutions:
The complete solution set is:

- \( x = \frac{\pi}{2} + n\pi \)
- \( x = \frac{\pi}{3} + 2n\pi \)
- \( x = \frac{5\pi}{3} + 2n\pi \)

Would you like more details on any part of this solution? Here are some related questions to consider:

1. What are the graphical representations of the solutions?
2. How do these solutions relate to the unit circle?
3. Can you explain the double angle identities in more detail?
4. What are the implications of these solutions in real-world scenarios?
5. How would you solve similar trigonometric equations? 

**Tip:** Always check for solutions within a specified interval, such as \( [0, 2\pi] \), to ensure completeness for specific applications.

Explore the solution to the trigonometric equation Cos(2x) - Cos(x) + 1 = 0 using double angle identities. This problem provides a comprehensive understanding of cosine functions and their properties, suitable for high school students.

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