Answer
$\frac{\pi}{8}$, $\frac{3\pi}{8}$, $\frac{5\pi}{8}$, $\frac{7\pi}{8}$, $\frac{9\pi}{8}$, $\frac{11\pi}{8}$, $\frac{13\pi}{8}$, $\frac{15\pi}{8}$
Work Step by Step
$\cos \theta \cos 3\theta-\sin \theta\sin 3\theta=0$
$\cos (\theta+3\theta)=0$
$\cos 4\theta=0$
$4\theta=\dots$, $\frac{\pi}{2}$, $\frac{3\pi}{2}$, $\frac{5\pi}{2}$, $\frac{7\pi}{2}$, $\frac{9\pi}{2}$, $\frac{11\pi}{2}$, $\frac{13\pi}{2}$, $\frac{15\pi}{2}$, $\dots$
Since we only want solutions in $[0, 2\pi)$, $\theta=\frac{\pi}{8}$, $\frac{3\pi}{8}$, $\frac{5\pi}{8}$, $\frac{7\pi}{8}$, $\frac{9\pi}{8}$, $\frac{11\pi}{8}$, $\frac{13\pi}{8}$, $\frac{15\pi}{8}$
