We are given the trigonometric equation:

\[
2 \cdot \cos^2(x) = \sin(2x)
\]

### Step 1: Use trigonometric identities

1. Recall the **double angle identity** for \(\sin(2x)\):

\[
\sin(2x) = 2 \sin(x) \cos(x)
\]

Substitute this identity into the original equation:

\[
2 \cdot \cos^2(x) = 2 \sin(x) \cos(x)
\]

### Step 2: Simplify the equation

We can divide both sides of the equation by 2:

\[
\cos^2(x) = \sin(x) \cos(x)
\]

### Step 3: Factor the equation

Move all terms to one side of the equation:

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

Factor out \(\cos(x)\):

\[
\cos(x) (\cos(x) - \sin(x)) = 0
\]

### Step 4: Solve for \(x\)

Now, solve the factored equation by considering two cases:

#### Case 1: \(\cos(x) = 0\)

\[
x = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}
\]

#### Case 2: \(\cos(x) - \sin(x) = 0\)

This simplifies to:

\[
\cos(x) = \sin(x)
\]

Divide both sides by \(\cos(x)\) (which is valid if \(\cos(x) \neq 0\)):

\[
1 = \tan(x)
\]

Thus,

\[
x = \frac{\pi}{4} + n\pi, \quad n \in \mathbb{Z}
\]

### Final Solution:

The general solution is:

\[
x = \frac{\pi}{2} + n\pi \quad \text{or} \quad x = \frac{\pi}{4} + n\pi, \quad n \in \mathbb{Z}
\]

Let me know if you want more details on any step.

### Related questions:

1. How do you derive the double angle identity for \(\sin(2x)\)?
2. What are other common trigonometric identities used in solving equations?
3. How do you graphically interpret the solutions to \(\cos(x) = \sin(x)\)?
4. Why is it important to factor equations when solving trigonometric identities?
5. What are the differences between general and specific solutions in trigonometry?

**Tip:** Always check the domain of the trigonometric functions when solving for \(x\).

Learn how to solve the trigonometric equation 2 cos^2(x) = sin(2x) with detailed step-by-step instructions. This problem covers important trigonometric concepts such as the double angle identity and solving factored trigonometric equations. Suitable for high school students in Grades 10-12.

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