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

We know that $\sin(2\alpha) = 2\sin(\alpha)\cos(\alpha)$ and $\cot(\alpha) = \frac{\cos(\alpha)}{\sin(\alpha)}$.

Substituting these into the expression, we get:

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

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

$= \frac{2\sin(\alpha)\cos^2(\alpha)}{2\sin(\alpha)}$

Assuming $\sin(\alpha) \neq 0$, we can cancel the $2\sin(\alpha)$ terms from the numerator and denominator.

$= \cos^2(\alpha)$

3. Final Answer

$\cos^2(\alpha)$