Question came from my math olympiad exercise book which doesn't have any solutions on the back of it, here is the problem
Prove that $\prod_{k=1}^{n}\cos\frac{2^{k}\pi}{2^{n}-1}=\frac{1}{2^{n}}$
Now here is what I did so far:
Since $2\sin x\cos x=\sin2x$ we can get $\prod_{k=1}^{n}\cos\frac{x}{2^{k-1}}=\frac{1}{2^{n}}\prod_{k=1}^{n}\frac{\sin\frac{2x}{2^{k-1}}}{\sin\frac{x}{2^{k-1}}}$
I'm kinda of stuck after that and don't really know what to do, any hints or solutions?
