The period of the function $f\left(x\right)={\cos }^{2}3x+\tan 4x$ is

$f\left(x\right)=(1/2)(1+\cos 6x)+\tan 4x$ The period of $\cos 6x\text{ is }2\pi /6=\pi /3$ and the period of tan $4x$is$\pi /4$. Hence

the period of f is$l.c.m.$ of $\pi /3$ and $\pi /4 = \pi$.