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for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.

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closed form for an alternating cosecant sum

Is there any closed form for the following finite sum $$\sum_{j=1}^{n-1}\frac{(-1)^j}{\sin (\frac{j\pi}{n})}$$ where $n$ is an even number? Any comment or reference is welcome.
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