What is the period of fx = cos(cos x) + cos(sin x)
The cos and sin functions both have period $\displaystyle 2\pi$, so it will certainly be true that $\displaystyle f(x+2\pi) = f(x).$ Thus f(x) repeats itself after an interval of $\displaystyle 2\pi.$ The only thing that you have to be wary about is whether that is the shortest interval after which f(x) repeats itself, or whether it might have a shorter period. Not wanting to give away the whole answer, I'll leave you to think about that.