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T. Amdeberhan
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modulus identity with roots of unity

Let $\omega_n=e^{\frac{\pi i}{2n+1}}$. I've an experimental encounter with certain relation involving roots of unity.

Question. Is this true? If yes, any proof? For $p\geq0$ an integer, we have the identity $$\sum_{j=1}^n\left\vert\frac{1-\omega_n^{2j}}{1+\omega_n^{2j}}\right\vert^{2p}= \sum_{j=1}^n\left\vert \frac{1+\omega_n^{2j-1}}{1-\omega_n^{2j-1}}\right\vert^{2p}.$$