Compute $\prod_{j=1}^{n-1}\sin\left(\frac{\pi j}{2 n}\right)$

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.

• It's the square root of $\prod_{j=1}^{2n-1}\sin(j\pi/2n)$. Can you do that one? – Lord Shark the Unknown Jun 27 '18 at 15:05
• @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 $j = n+1 \to 2n-1$ – John Lou Jun 27 '18 at 15:40