Confusion in finding Period of function

Question

My instructor said this

If $f$ has period $T_1$ and $g$ has period $T_2$, then $\frac{f}{g}$ has period which is least common multiple of $T_1$ and $T_2$.

But if I take $f=\sin(x)$ and $g = \cos(x)$ then above rule fails. Why is this?

Thanks

Answer 1

As dxiv said in the comments, the rule will only give a period, and not the minimal period of the function $f/g$. To take a trivial example, let $f=g=\sin(x),$ then $LCM(per(f),per(g))=2\pi,$ but $f/g = 1$ which has period $0.$