How can I show analytically (easy way) that the function $f(x) = \cos(x)-\cos(2x)$ has a period of $2\pi$ ?

I can find the period solving $f(x)=f(x+T)$. 

$\cos(x)-\cos(2x) = \cos(x+T)-\cos(2x+2T)$. If $T=2\pi$ then:

$\cos(x)-\cos(2x) = \cos(x+2\pi)-\cos(2x+4\pi)$

This last statement is true. 

The demonstration can be done in this way?

thanks in advance