<s>[INST] Is $\cos(4x)+\sin(2x)$ periodic and how do I calculate the primitive period? My first attempt is under this, i can work out the primitive period of both of the $\cos(4x)$ and $\sin(2x)$ but how do I calculate the primitive period of $\cos(4x)+\sin(2x)$?
My attempt: 
Let $u=4x$ then $x=\frac{u}{4}$ and $\cos(u)$ is $2\pi$ periodic thus $T=\frac{2\pi}{4}$ hence $\cos(4x)$ is periodic with primitive period $T=\frac{\pi}{4}$.
Now Let $H=2x$ and thus $x=\frac{H}{2}$ and $\sin(H)$ is also $2\pi$ periodic Thus $\sin(2x)$ is periodic with primitive period $T=\frac{2\pi}{2}=\pi$ but i dont know how to combine these results to calculate the primitive period of the sum of both $\cos(4x)$ and $\sin(2x)$
 [/INST] Your attempt seems fine. Choose $T = \pi$, because that is a multiple of $T_1 = \pi /4$ and $T_2 = \pi$. Then, \begin{align} f(x + T) & = \cos (4(x+T)) + \sin (2(x+T)) \\ & = \cos (4x + 4T) + \sin (2x + 2T) \\ & = \cos(4x+4\pi) + \sin (2x + 2 \pi) \\ & = \cos (4x) + \sin (2x) \\ & = f(x). \end{align}
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