Find the general solutions of cos 2x = 1.

Answer

To find the general solutions of the equation \( \cos 2x = 1 \), we can follow these steps:

### Step 1: Understand the equation
We start with the equation:
\[ \cos 2x = 1 \]

### Step 2: Identify the angles for which cosine equals 1
The cosine function equals 1 at specific angles. The general solution for \( \cos \theta = 1 \) is given by:
\[ \theta = 2n\pi \quad \text{where } n \text{ is any integer} \]
Thus, we can set:
\[ 2x = 2n\pi \]

### Step 3: Solve for \( x \)
To isolate \( x \), we divide both sides of the equation by 2:
\[ x = n\pi \]

### Step 4: Write the general solution
The general solution for the equation \( \cos 2x = 1 \) is:
\[ x = n\pi \quad \text{where } n \text{ is any integer} \]

### Final Answer:
The general solutions of \( \cos 2x = 1 \) are:
\[ x = n\pi \quad (n \in \mathbb{Z}) \]

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