How to prove that $\sum_{ k= 1}^{N-1} \sin(\frac{k}{N} \pi)^{-2} = (N^2 - 1)/3$?

Question

I find the equality \begin{equation} \sum_{k = 1}^{N-1} \sin(\frac{k}{N}\pi)^{-2} = \frac{N^2 - 1}{3}, \end{equation} during study.

Just wonder how can I prove it?

Short matlab code to do the verification:

N = 10;
xv = (1:N-1)/N*pi; 
error = sum(sin(xv).^(-2)) - (N^2 -1)/3

EDIT2: I found the proof of $\Pi_{k = 1}^{N-1} \sin(\frac{k \pi}{N}) = \frac{N}{2^{N-1}}$ here it may be helpful.

Thank you very much!