Are sums ${ \sum _{j=1} ^{n-1} \frac{1}{\sin( \frac{2 \pi}{2n} j) ^{2m} } }$ rational?

Question

[Reposted from this tweet by Prof. Samuel Walters]

Proposition: For integers ${ m \geq 1, n \geq 2 }$ the sum ${ \sum _{j=1} ^{n-1} \frac{1}{\sin \left( \frac{2 \pi}{\color{purple}{2n}} j \right) ^{\color{purple}{2m}} } }$ is a rational number of the form ${ \left(\frac{2}{n} \right) ^m \cdot (\text{an integer } Z(m,n)). }$

As mentioned, the integers ${ Z(m,n) }$ ".. arise from a special case of a formula of E. Verlinde for the dimension of the zeroth cohomology group of some moduli space. See pg 177 in: E. Witten, On Quantum Gauge Theories in Two Dimensions".

Is there an elementary proof of this (that the sums are rational) ?

Thanks in advance !