If $N$ is odd, is $\sum_{k=1}^{(N-1)/2}\frac{1}{\sin^2(\pi k/N)}$ always rational?

Question

For an odd number of $N$, derive the following

$$\sum_{k=1}^{(N-1)/2}\frac{1}{\sin^2(\pi k/N)}$$

Numerically, it seems to be a rational number, but I can't prove it.

• For $N=3$, the answer is $4/3$.
• For $N=5$, the answer is $4$.
• For $N=7$, the answer is $8$.
• For $N=9$, the answer is $40/3$.
• For $N=101$, the answer is $1700$.