$\begin{aligned}
& \cos 2 \theta \cos \frac{\theta}{2}+\cos \frac{5 \theta}{2}=2 \cos ^3 \frac{5 \theta}{2} \
& \frac{1}{2}\left(2 \cos 2 \theta \cos \frac{\theta}{2}\right)+\cos \frac{50}{2} \
& =\frac{1}{2}\left(\cos \frac{15 \theta}{2}+3 \cos \frac{5 \theta}{2}\right)
\end{aligned}$
or solving
$\begin{aligned}
& \cos \frac{3 \theta}{2}=\cos \frac{15 \theta}{2} \
& \cos \frac{15 \theta}{2}-\cos \frac{3 \theta}{2}=0 \
& 2 \sin 30 \sin \frac{9 \theta}{2}=0
\end{aligned}$
3θ=nπ or 29θ=mπ
$\begin{aligned}
& \theta=\frac{\mathrm{n} \pi}{3} \quad \theta=\frac{2 \mathrm{~m} \pi}{9} \
& \theta=\left{-\frac{\pi}{2}, \frac{\pi}{3}, 0\right} \
& \theta=\left{-\frac{4 \pi}{9}, \frac{-2 \pi}{9}, \frac{4 \pi}{9}, \frac{2 \pi}{9}\right}
\end{aligned}$