How can I evaluate $\prod_{j=1}^{n-1}\sin\left(\frac{\pi j}{2 n}\right)$ ? I know that without the factor $2$ one can take advantage of the roots of the unity, but in this case only the roots with positive imaginary parts are to be considered.

Thanks in advance.

  • 4
    $\begingroup$ It's the square root of $\prod_{j=1}^{2n-1}\sin(j\pi/2n)$. Can you do that one? $\endgroup$ – Lord Shark the Unknown Jun 27 '18 at 15:05
  • $\begingroup$ I can solve the productory you write, though it's not obvious to me that my expression is the square root of this one. $\endgroup$ – Graz Jun 27 '18 at 15:32
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    $\begingroup$ @GrazianoAmati Notice that the sin values are reflected across $\pi/2$. Therefore, the product from $j = 1 \to n-1$ is the same as the product from $j = n+1 \to 2n-1$ $\endgroup$ – John Lou Jun 27 '18 at 15:40
  • $\begingroup$ Try to follow this technique math.stackexchange.com/questions/2649011/… $\endgroup$ – rtybase Jun 27 '18 at 15:56

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