\prod_{k=1}^{17}\cos\left(\frac{k\pi}{17}\right)=\cos \pi \prod_{k=1}^{16}\cos\left(\frac{k\pi}{17}\right)=-(2i)^{-16}\frac{1-(-1)^{17}}{2(-1)^{16}}=-\frac{1}{2^{16}}=-\frac{1}{65536}