To solve the equation \( \cos(x) + \cos(2x) = 0 \), follow these steps:

### Step 1: Use the double-angle identity
Recall that the double-angle identity for cosine is:
\[
\cos(2x) = 2\cos^2(x) - 1
\]
Substitute this into the equation:
\[
\cos(x) + 2\cos^2(x) - 1 = 0
\]

### Step 2: Rearrange the equation
Rearrange the equation into a standard quadratic form:
\[
2\cos^2(x) + \cos(x) - 1 = 0
\]

### Step 3: Solve the quadratic equation
Let \( y = \cos(x) \). The equation becomes:
\[
2y^2 + y - 1 = 0
\]
Now solve this quadratic equation using the quadratic formula:
\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 2 \), \( b = 1 \), and \( c = -1 \). Plugging in these values:
\[
y = \frac{-1 \pm \sqrt{1^2 - 4(2)(-1)}}{2(2)} = \frac{-1 \pm \sqrt{1 + 8}}{4} = \frac{-1 \pm \sqrt{9}}{4} = \frac{-1 \pm 3}{4}
\]
This gives us two solutions:
\[
y = \frac{-1 + 3}{4} = \frac{2}{4} = \frac{1}{2} 
\]
\[
y = \frac{-1 - 3}{4} = \frac{-4}{4} = -1
\]
So, \( \cos(x) = \frac{1}{2} \) or \( \cos(x) = -1 \).

### Step 4: Find the corresponding angles
1. \( \cos(x) = \frac{1}{2} \) gives:
   \[
   x = \pm \frac{\pi}{3} + 2k\pi, \quad k \in \mathbb{Z}
   \]
2. \( \cos(x) = -1 \) gives:
   \[
   x = \pi + 2k\pi, \quad k \in \mathbb{Z}
   \]

### Final Answer:
The general solutions are:
\[
x = \frac{\pi}{3} + 2k\pi, \quad x = -\frac{\pi}{3} + 2k\pi, \quad x = \pi + 2k\pi, \quad k \in \mathbb{Z}
\]

Would you like more details on any part of this solution? Here are five related questions you might find interesting:

1. How do you derive the double-angle identities for cosine and sine?
2. What are other forms of trigonometric identities that could simplify trigonometric equations?
3. How can you solve trigonometric equations involving sine or tangent functions?
4. What are the implications of solving trigonometric equations over different intervals?
5. How does the general solution for trigonometric equations change if a different trigonometric function is involved?

**Tip:** When solving trigonometric equations, always check if identities can simplify the problem before applying algebraic methods.

Learn how to solve the trigonometric equation cos(x) + cos(2x) = 0 step-by-step. This problem involves using the double-angle identity for cosine and solving a resulting quadratic equation. Suitable for advanced high school students and college-level trigonometry courses.

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