A friend showed me this problem, I'm not sure where it comes from but probably from some signal processing background.
Let $g_k(\phi)=\max_t|\cos(t)+\cos(kt+\phi)|$ for $\phi\in[0,2\pi[$. Then it seem to be the case that for $k$ an odd integer, $\min_\phi g_k(\phi)=g_k(\pi)$and for $k$ an even integer $\min_\phi g_k(\phi)=g_k(\pi/2)$.
In general it would be interesting to know $\textrm{argmin}_\phi g_k(\phi)$ for any $k$ (I think we may have to chance the $\max$ to a $\sup$ when $k$ is irrational).