could you help me with the following problem or at least give some ideas?
Let $F: \mathbb{R} \to \mathbb{C}$ be a polynomial of degree $2n$ defined as follows: $$F(x)=(x+i)^{2n+1}-(x-i)^{2n+1}$$
Now, let $\alpha_1,\alpha_2,...,\alpha_n \in (0,\frac{\pi}{2})$ be the distinct roots of the equation $$F(cotx)=0$$. The task is to find the sum $$\sum_{k=1}^n (cot(\alpha_k))^2$$
This is a homework problem from the complex analysis course which should be solved using the Residue Theorem.
We also have a hint that there is a polynomial $W$ of degree $n$ such that $W(x^2)=F(x)$.