# How do you prove tan(x/2)= sinx+cosxcotx-cotx?

We know that $\tan \left(\frac{x}{2}\right) = \frac{1 - \cos \left(x\right)}{\sin} \left(x\right)$. So we develop the right side of the equality. $\cot \left(x\right) = \frac{1}{\tan} \left(x\right)$ so :
$\sin \left(x\right) + \cos \left(x\right) \cot \left(x\right) - \cot \left(x\right) = \frac{{\sin}^{2} \left(x\right) + {\cos}^{2} \left(x\right) - \cos \left(x\right)}{\sin} \left(x\right) = \frac{1 - \cos \left(x\right)}{\sin} \left(x\right) = \tan \left(\frac{x}{2}\right)$.