I am interested in the sum $$ \sum_{n=1}^k 2\biggl[\sin\biggl(\frac{n\pi}{2k+2}\biggr)\biggr]^{-2g} $$ where $k,g$ are integers. It is not too hard to show that this can also be expressed as $$ -1-2\pi i\underset{t=0}{\operatorname{Res}}\biggl[\sin\biggl(\frac{\pi t}{2k+2}\biggr)\biggr]^{-2g}\frac{1}{e^{2\pi i t}-1} $$

I also want the sum $$ \sum_{n=1}^k 2(-1)^n\biggl[\sin\biggl(\frac{n\pi}{2k+2}\biggr)\biggr]^{-2g} $$ but this one -- I have not been able to express in a simpler form. Is there a representation similar to the previous one? Or even better, an even simpler representation, for both (not necessarily using residues)?

I am interested in the sum $$ \sum_{n=1}^k 2\biggl[\sin\biggl(\frac{n\pi}{2k+2}\biggr)\biggr]^{-2g} $$ where $k$, $g$ are integers. It is not too hard to show that this can also be expressed as $$ -1-2\pi i\underset{t=0}{\operatorname{Res}}\biggl[\sin\biggl(\frac{\pi t}{2k+2}\biggr)\biggr]^{-2g}\frac{1}{e^{2\pi i t}-1}. $$

I also want the sum $$ \sum_{n=1}^k 2(-1)^n\biggl[\sin\biggl(\frac{n\pi}{2k+2}\biggr)\biggr]^{-2g} $$ but this one — I have not been able to express in a simpler form. Is there a representation similar to the previous one? Or even better, an even simpler representation, for both (not necessarily using residues)?