How to prove periodicity of a trigonometric function 
$f(x)= \sin(2x)+3\cos(8x)$
Is the function periodic ?

What I did is equalize $f(x)=f(x+T)$ and after noting that $\sin(2x)=\sin(2T)=\sin(8x)=0$ and $\cos(2x)=\cos(2T)=\cos(8x)=1$ we get that both sides of the equation equal 3.
Is that enough to show that the function is periodic ?
Thanks.

no need to be too complicated here:
both $sin \;x$ and $cos \;x$ are perodic, each with period $2\pi$
so $sin \; 2x$ is periodic, with period $\frac{2\pi}{2} = \pi$
and $cos \; 8x$ is periodic, with period $\frac{2\pi}{8} = \frac{\pi}4$
can you find the least common multiple of $\pi$ and $\frac{\pi}4$?
do you see why that must be the period of $\sin(2x)+3\cos(8x)$?
