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.

$$\begin{align}f(x+\pi)&=\sin(2(x+\pi))+3\cos(8(x+\pi))\\
&=\sin(2x+2\pi)+3\cos(8x+8\pi)\\
&=\sin(2x)\cos(2\pi)+\cos(2x)\sin(2\pi)+3\cos(8x)\cos(8\pi)-3\sin(8x)\sin(8\pi)\\
&=\sin(2x)+3\cos(8x)\\
&=f(x)\end{align}$$
