Originally Posted by

**Soroban**

$\displaystyle [1]\;\;4\cos(3x) \:=\:0\quad\Rightarrow\quad 3x \:=\:\frac{\pi}{2},\:\frac{3\pi}{2},\:\frac{5\pi}{ 2},\:\hdots \quad\Rightarrow\quad\boxed{ x \;=\;\frac{\pi}{6},\:\frac{\pi}{2},\:\frac{5\pi}{6 },\:\hdots}$

$\displaystyle [2]\;\;\cos^2(3x) \:=\:1\quad\Rightarrow\quad \cos3x \:=\:\pm1\quad\Rightarrow\quad 3x \;=\;0,\:\pi,\:2\pi,\:3\pi,\:4\pi,\:\hdots $

. . . $\displaystyle \boxed{x \;=\;0,\:\frac{\pi}{3},\:\frac{2\pi}{3},\:\pi,\:\f rac{4\pi}{3},\:\hdots} $

It is always good to check the answers.

Often, some are extraneous roots.

. . (No, I didn't check these . . .)