<s>[INST] Show that the function $f(x)=\cos x+\cos\left(\frac{\sqrt{3}}{2}x\right)$ is not periodic. Question: Show that the function $f(x)=\cos x+\cos\left(\frac{\sqrt{3}}{2}x\right)$ is not periodic. 
My approach: Let us assume for the sake of contradiction that $T>0$ be the period of $f$. Then we must have $\forall a\in\mathbb{R}$ $$\int_a^{a+T}f(x)dx=\int_0^Tf(x)dx \hspace{1cm}...(1)$$ 
Therefore by setting $a=2\pi$ in $(1)$ we have, $$2\cos\left(\sqrt{3}\pi+\frac{\sqrt{3}}{4}T\right)\sin\frac{\sqrt{3}}{4}T=2\sin\frac{\sqrt{3}}{4}T\cos\frac{\sqrt{3}}{4}T \\\implies\sin\frac{\sqrt{3}}{4}T\left(\cos\left(\sqrt{3}\pi+\frac{\sqrt{3}}{4}T\right)-\cos\frac{\sqrt{3}}{4}T\right)=0$$
$$\implies \sin\frac{\sqrt{3}}{4}T=0, \cos\left(\sqrt{3}\pi+\frac{\sqrt{3}}{4}T\right)-\cos\frac{\sqrt{3}}{4}T=0.$$ This implies that either $T=\frac{4}{\sqrt 3}n\pi$ for some $n\in\mathbb{N}$ or $T=\left(\frac{4}{\sqrt 3}m-2\right)\pi$ for some $m\in\mathbb{N}$. 
How to proceed after this?
 [/INST] $$f(x)=\cos x+\cos\frac{\sqrt{3} x}{2}$$
Let $f_1(x)$ be periodic with a period of $T_1$ and $f_2(x)$ be periodic with a period of $T_2$. Then $f(x)=f_1(x)+f_2(x)$ only if $T_1/T_2$ is rational. Then the period is given by $T=LCM(T_1,T_2).$
Here $T_1=2\pi$ and $T_2=\frac{4\pi}{\sqrt{3}}$ This means $T_1/T_2=\frac{\sqrt{3}}{2}$, it being irrational the given  $f(x)$ is not periodic. Equivalently, here the LCM of $T_1$ and $T_2$ does not exist.
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