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