\frac{1}{2}\tan{\frac{x}{2}}+\dots +\frac{1}{2^n}\tan^n{\frac{x}{2^n}}+\frac{1}{2^{n+1}}\tan^{n+1}{\frac{x}{2^{n+1}}}
=\frac{1}{2^{n+1}}\cot{\frac{x}{2^{n+1}}}-\cot{x}.