Let $n$ be a nonnegative integer. Can you help me prove the following ?
There exists a unique polynomial $P_{n}$ such that for all $t \in [0,\frac{\pi}{2}]$, $P_{n}(\operatorname{cotan}^2t)=\frac{\sin((2n+1)t}{(\sin t)^{2n+1}}$ with $\operatorname{cotan}(x)=\frac{\cos x}{\sin x}$.