Quasi-Periodic Function as the Sum of Periodic Functions

Question

Define $$f(x)=\frac{\cos^2(\pi x)}{2+\cos(x)}$$ We know that $f(x)$ is not periodic. Is there any way to write $f(x)$ as the sum of two periodic functions. That is, find periodic functions $f_1(x)$ and $f_2(x)$ such that $f(x)=f_1(x)+f_2(x)$.

Answer 1

No. Suppose $f=f_1+f_2$, where $f_1$ has period $p_1$ and $f_2$ has period $p_2$. Then $$f(x+p_1)=f_1(x+p_1)+f_2(x+p_1) = f_1(x)+f_2(x+p_1),$$ and $$f(x+p_1+p_2)=f_1(x+p_2)+f_2(x+p_1);$$ similarly, $$f(x+p_2)=f_1(x+p_2)+f_2(x),$$ for all $x$. Combining these, we see that for all $x$, we have $$ f(x+ p_1+p_2)-f(x+p_2)-f(x+p_1)+f(x)=0.\tag{*}$$

After clearing fractions in (*) you get a non-trivial trigonometric polynomial that vanishes for all values of $x$.