The problem asks to simplify the expression $\sin(2\alpha) \cdot \frac{\cot(\alpha)}{2}$.

1. Problem Description

The problem asks to simplify the expression $\sin(2\alpha) \cdot \frac{\cot(\alpha)}{2}$.

2. Solution Steps

First, we can express $\sin(2\alpha)$ in terms of $\sin(\alpha)$ and $\cos(\alpha)$ using the double angle identity:

$\sin(2\alpha) = 2\sin(\alpha)\cos(\alpha)$

Next, we can express $\cot(\alpha)$ in terms of $\sin(\alpha)$ and $\cos(\alpha)$:

$\cot(\alpha) = \frac{\cos(\alpha)}{\sin(\alpha)}$

Now, substitute these expressions into the original equation:

$2\sin(\alpha)\cos(\alpha) \cdot \frac{\frac{\cos(\alpha)}{\sin(\alpha)}}{2}$

Simplify the expression:

$2\sin(\alpha)\cos(\alpha) \cdot \frac{\cos(\alpha)}{2\sin(\alpha)}$

We can cancel out the common factors $2$ and $\sin(\alpha)$ (assuming $\sin(\alpha) \ne 0$):

$\cos(\alpha) \cdot \cos(\alpha)$

This simplifies to:

$\cos^2(\alpha)$

3. Final Answer

$\cos^2(\alpha)$