Is there an explicit formula for the inverse of $\cot\left(\frac{x}{2}\right)\sqrt{1-\cos(x)}$?

Question

I apologize if this is trivial but I am stuck.

Given the bijective function $f:(0,2\pi) \to (-2,2)$ with $$ f(x)=\cot\left(\frac{x}{2}\right)\sqrt{1-\cos(x)} $$ where $\cot$ is the cotangent, how can I find an inverse $g:(-2,2)\to (0,2\pi)$? Is there an explicit formula?

Answer 1

Hint: Use this fact that $$\frac{1-\cos(x)}{2}=\sin^2(x/2)$$ and that if $x\in(0,2\pi), |\sin(x/2)|=\sin(x/2)$