I've proved that $\cos(\cos(x))$ restricted to $[0,\frac \pi 2]$ is a contraction, which imply by Banach's fixed point theorem that it has a unique fixed point on this interval.
I've also proved that $\cos(x)$ must also have a unique fixed point on this interval, and that it must be the same as the one for $\cos(\cos(x))$.
But, how can I find this fixed point explicitly ?