Answer
$0$, $\frac{\pi}{3}$, $\frac{2\pi}{3}$, $\pi$, $\frac{4\pi}{3}$, $\frac{5\pi}{3}$
Work Step by Step
$\cos 2\theta-\cos 4\theta=0$
$\cos 2\theta-\cos (2*2\theta)=0$
$\cos 2\theta-(2\cos^2 2\theta-1)=0$
$\cos 2\theta-2\cos^2 2\theta+1=0$
$2\cos^2 2\theta-\cos 2\theta-1=0$
$(2\cos 2\theta+1)(\cos 2\theta-1)=0$
If $2\cos 2\theta+1=0$, then $2\cos 2\theta=-1$, and $\cos 2\theta=-\frac{1}{2}$. Then $2\theta=\dots$, $\frac{2\pi}{3}$, $\frac{4\pi}{3}$, $\frac{8\pi}{3}$, $\frac{10\pi}{3}$, $\dots$, which means the only values for $\theta$ in $[0, 2\pi)$ are $\frac{\pi}{3}$, $\frac{2\pi}{3}$, $\frac{4\pi}{3}$, and $\frac{5\pi}{3}$.
If $\cos 2\theta-1=0$, then $\cos 2\theta=1$, and $2\theta=\dots$, $0$, $2\pi$, $\dots$. The only values for $\theta$ in $[0, 2\pi)$ are $0$ and $\pi$.
