Representations of sums of even negative degrees of sines at equally spaced points (Q1357947)

In this interesting paper, the author studies the following trigonometric sums \[ S_n(N)= \sum^{N- 1}_{k=1}\sin^{- 2n}{\pi k\over N} \] for positive integers, \(n\) and \(N\), where \(N\geq 2\). Among others, he shows that for any positive integer \(n\) the sum \(S_n(N)\) is an algebraic polynomial of order \(n\) in variable \(N^2\), and this polynomial has divisor \(N^2- 1\), e.g., \(S_2(N)={1\over 45}(N^2- 1)(N^2+ 11)\).