While trying to prove some identities for generating functions, I ended up needing to show that

$$\sum_{p=1}^n \exp\left(\frac{i\pi p l}{2m}\right)\prod_{\substack{k=1\\k\neq p}}^n\frac{1}{\sin\left(\frac{\pi (k-p)}{2m}\right)} \stackrel{?}{=} 0$$

for integers $m \geq 1$, $2\leq n\leq 2m$, and $l = -n+2,-n+4,\ldots n-2$. Is this identity known? I have checked it to be valid for small values, but so far I have been unable to prove the general case.