Let $m\geq 2$ be an integer, then there is the well known formula $$\sum\limits_{j=1}^{m-1}\frac{1}{\sin^2(\frac{j\pi}{m})}=\frac{m^2-1}{3},$$
I'm interested in similar equations for the following expression
$$\sum\limits_{j=1}^{m-1}\frac{1}{\sin^{2p}(\frac{j\pi}{m})}=(...),$$
for any $p\in\mathbb{N}$. Does someone know whether there are well known formulas like for $p=1$?
Best wishes
Your sums may be seen (with a minor fix) as $$ \sum_{\zeta\in Z} \zeta^{-2p} $$ where $Z$ is the set given by the roots of a Chebyshev polynomial of the second kind. By Newton's identities, in order to compute our power sums it is enough to compute the right number of coefficients of a Chebyshev polynomial of the second kind. In order to do that, we just need to exploit the binomial theorem, giving: $$ U_n(x)=\sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}\binom{n-k}{k}(-1)^k (2x)^{n-2k}.$$