Find the general solution : `cos 4x = cos 2x`

Explanation

To solve the equation \( \cos 4x = \cos 2x \), we can follow these steps:

### Step 1: Use the cosine identity
We know that if \( \cos A = \cos B \), then:
\[ A = 2n\pi \pm B \quad \text{for } n \in \mathbb{Z} \]
In our case, let \( A = 4x \) and \( B = 2x \). Therefore, we can write:
\[ 4x = 2n\pi \pm 2x \]

### Step 2: Solve the two cases
We will consider both cases separately.

#### Case 1: \( 4x = 2n\pi + 2x \)
Subtract \( 2x \) from both sides:
\[ 4x - 2x = 2n\pi \]
\[ 2x = 2n\pi \]
Dividing both sides by 2 gives:
\[ x = n\pi \]

#### Case 2: \( 4x = 2n\pi - 2x \)
Add \( 2x \) to both sides:
\[ 4x + 2x = 2n\pi \]
\[ 6x = 2n\pi \]
Dividing both sides by 6 gives:
\[ x = \frac{n\pi}{3} \]

### Step 3: Combine the solutions
From both cases, we have the general solutions:
1. \( x = n\pi \)
2. \( x = \frac{n\pi}{3} \)

### Final General Solution
Thus, the general solution for the equation \( \cos 4x = \cos 2x \) is:
\[ x = n\pi \quad \text{and} \quad x = \frac{n\pi}{3} \quad \text{for } n \in \mathbb{Z} \]

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