Answer
$\frac{\pi}{2}+k\pi, \frac{\pi}{9}+\frac{2k\pi}{3}, \frac{5\pi}{9}+\frac{2k\pi}{3}$
Work Step by Step
Use the Sum-to-Product Formula $\cos x+\cos y=2\cos\frac{x+y}{2}\cos\frac{x-y}{2}$.
$\cos 4\theta+\cos 2\theta=\cos \theta$
$2\cos\frac{4\theta+2\theta}{2}\cos\frac{4\theta-2\theta}{2}=\cos \theta$
$2\cos 3\theta\cos\theta=\cos \theta$
$2\cos 3\theta\cos\theta-\cos \theta=0$
$\cos\theta(2\cos 3\theta-1)=0$
If $\cos\theta=0$, then $\theta=\frac{\pi}{2}+k\pi$.
If $2\cos 3\theta-1=0$:
$2\cos3\theta=1$
$\cos 3\theta=\frac{1}{2}$
$3\theta=\frac{\pi}{3}+2k\pi, \frac{5\pi}{3}+2k\pi$
$\theta=\frac{\pi}{9}+\frac{2k\pi}{3}, \frac{5\pi}{9}+\frac{2k\pi}{3}$
