\prod_{k=1}^{n-1}\sin(k\pi/n)=\prod_{k=1}^{n-1}\left(\frac{\exp(ik\pi/n)-\exp(-ik\pi/n)}{2i}\right)=(2i)^{1-n} \exp(-i\pi (n-1)/2)\prod_{k=1}^{n-1}\left(\exp(2ik\pi/n)-1\right)=2^{1-n} \prod_{k=1}^{n-1} (1-\exp(2ik\pi/n))