Find the general solution : `cos 3x + cos x cos 2x = 0`

Explanation

To solve the equation \( \cos 3x + \cos x \cos 2x = 0 \), we will follow these steps:

### Step 1: Rewrite the equation using trigonometric identities
We know that:
\[ \cos 3x = 4\cos^3 x - 3\cos x \]
and
\[ \cos 2x = 2\cos^2 x - 1. \]
Substituting these identities into the equation gives:
\[ 4\cos^3 x - 3\cos x + \cos x(2\cos^2 x - 1) = 0. \]

### Step 2: Simplify the equation
Now, we can expand and simplify:
\[ 4\cos^3 x - 3\cos x + 2\cos^3 x - \cos x = 0. \]
Combining like terms results in:
\[ (4\cos^3 x + 2\cos^3 x) - (3\cos x + \cos x) = 0, \]
which simplifies to:
\[ 6\cos^3 x - 4\cos x = 0. \]

### Step 3: Factor the equation
We can factor out \( \cos x \):
\[ \cos x(6\cos^2 x - 4) = 0. \]

### Step 4: Set each factor to zero
This gives us two equations to solve:
1. \( \cos x = 0 \)
2. \( 6\cos^2 x - 4 = 0 \)

### Step 5: Solve the first equation
For \( \cos x = 0 \):
\[ x = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}. \]

### Step 6: Solve the second equation
For \( 6\cos^2 x - 4 = 0 \):
\[ 6\cos^2 x = 4 \implies \cos^2 x = \frac{2}{3} \implies \cos x = \pm \sqrt{\frac{2}{3}}. \]

### Step 7: Find the general solutions for \( \cos x = \sqrt{\frac{2}{3}} \) and \( \cos x = -\sqrt{\frac{2}{3}} \)
1. For \( \cos x = \sqrt{\frac{2}{3}} \):
\[ x = \cos^{-1}\left(\sqrt{\frac{2}{3}}\right) + 2n\pi \quad \text{or} \quad x = -\cos^{-1}\left(\sqrt{\frac{2}{3}}\right) + 2n\pi, \quad n \in \mathbb{Z}. \]

2. For \( \cos x = -\sqrt{\frac{2}{3}} \):
\[ x = \cos^{-1}\left(-\sqrt{\frac{2}{3}}\right) + 2n\pi \quad \text{or} \quad x = -\cos^{-1}\left(-\sqrt{\frac{2}{3}}\right) + 2n\pi, \quad n \in \mathbb{Z}. \]

### Final General Solution
Combining all solutions, we have:
1. \( x = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z} \)
2. \( x = \cos^{-1}\left(\sqrt{\frac{2}{3}}\right) + 2n\pi \)
3. \( x = -\cos^{-1}\left(\sqrt{\frac{2}{3}}\right) + 2n\pi \)
4. \( x = \cos^{-1}\left(-\sqrt{\frac{2}{3}}\right) + 2n\pi \)
5. \( x = -\cos^{-1}\left(-\sqrt{\frac{2}{3}}\right) + 2n\pi \)