Half-Angle Identity

How would you do from $\frac{\sin \theta}{2-2 \cos \theta}$ to $\frac{1}{2} \cot \left( \frac{ \theta }{2} \right)$ ?
I tried it for a while but the squaring in the $\cos$ identity is causing confusion. Thank in advance.
$\cot \frac{\theta}{2} = \pm\, \sqrt\frac{1 + \cos \theta}{1 - \cos \theta} = \frac{1 + \cos \theta}{\sin \theta} = \frac{\sin \theta}{1 - \cos \theta} = \csc \theta + \cot \theta.\,$