I understand that $ f(x)=\cos(x)\cos(3x) $ is clearly periodic with a period of $T= 2\pi$ ($ T_1=2\pi, T_2=2\pi/3 ; T=3T_2=T_1=2\pi$). This is how I've learned to find a period of any $f(x)$ that consists of a sum/product of two functions... But is it actually useless when finding a fundamental period?
It's quite clear that $2\pi$ is a period of $f(x)$, but I can't understand how I can find the fundamental period of this function. The solution is supposed to be $\pi$ and it's obvious since $f(x+\pi)=f(x)$ in this case, but how do I find it algebraically?
I've searched through numerous answers and they wonderfully explain what a fundamental period actually is, but I understand that. I just don't know how to find it myself.
This is another way to do it if you know what is the fundamental period of $\cos^2(x)$.
You have $f(x)=\cos(x) \cdot \cos(3x)=\cos^2(x)(4\cos^2(x)-3)$
How would you justify the fact that the fundamental period of $\cos^2(x)$ is $\pi$?