Simplify the expression $\sin(2\alpha) \cdot \frac{\cot(\alpha)}{2}$.

1. Problem Description

Simplify the expression $\sin(2\alpha) \cdot \frac{\cot(\alpha)}{2}$.

2. Solution Steps

We start with the given expression:

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

We can use the double angle formula for sine:

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

Substituting this into the expression, we get:

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

We can simplify by canceling the 2 in the numerator and denominator:

$\sin(\alpha)\cos(\alpha) \cdot \cot(\alpha)$

We also know that $\cot(\alpha) = \frac{\cos(\alpha)}{\sin(\alpha)}$, so we can substitute this:

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

Now, we can cancel the $\sin(\alpha)$ terms:

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

This simplifies to:

$\cos^2(\alpha)$

3. Final Answer

$\cos^2(\alpha)$