Answer
$\frac{\pi}{3}$, $\frac{5\pi}{3}$
Work Step by Step
$\cos \theta\cos 2\theta+\sin \theta\sin 2\theta=\frac{1}{2}$
$\cos (\theta-2\theta)=\frac{1}{2}$
$\cos (-\theta)=\frac{1}{2}$
Since cosine is an even function, $\cos \theta=\frac{1}{2}$.
The only solutions in $[0, 2\pi)$ are $\frac{\pi}{3}$ and $\frac{5\pi}{3}$.
