Find all $1<k,n,m\in\mathbb N$ for which $\cos^k(\frac{2\pi}n)=\cos(\frac{2\pi}m).$
The case where $k=2$ has been solved in my previous question.
I've not been able to find any new cases for $k>3$.
We know that $\mathbb Q\to\mathbb Q\left[\cos\left(\frac{2\pi}{m}\right)\right]\to \mathbb Q\left[\root k \of{\cos\left(\frac{2\pi}{m}\right)}\right]$ is of degree $k\varphi(m)$. So, $k\varphi(m)=\varphi(n)$.