Let $x = \pi/(2k+1)$, for $k>0$.
Prove that

$$
\cos(x)\cos(2x)\cos(3x)\dots\cos(kx) = \frac{1}{2^k}
$$

I've confirmed this numerically for $n$ from $1$ to $30$. I'm finding it surprisingly difficult using standard trigonometric formula manipulation. Even for the case $k = 2$, I needed to actually work out $\cos x$ by other methods to get the result.

Please let me know if you have a neat proof.