$2 \cos 2 \theta \cos \frac{\theta}{2}=2 \cos 3 \theta \cos \frac{9 \theta}{2}$ $\begin{aligned} & \cos \frac{5 \theta}{2}+\cos \frac{3 \theta}{2}=\cos \frac{15 \theta}{2}+\cos \frac{3 \theta}{2} \\\\ & \cos \frac{5 \theta}{2}=\cos \frac{15 \theta}{2} \\\\ & \frac{15 \theta}{2}=2 n \pi \pm \frac{5 \theta}{2} \\\\ & \frac{15 \theta}{2} \pm \frac{5 \theta}{2}=2 n \pi \\\\ & 10 \theta=2 n \pi \quad \text { or } 5 \theta=2 n \pi \\\\ & \theta=\frac{n \pi}{5} \text { or } \theta=\frac{2 n \pi}{5} \\\\ & \Rightarrow \theta=\frac{n \pi}{5} \end{aligned}$ $\begin{aligned} & \theta=\pm \pi, \pm \frac{4 \pi}{5}, \pm \frac{3 \pi}{5}, \pm \frac{2 \pi}{5}, \pm \frac{\pi}{5} \\\\ & m=5, \quad n=5 \\\\ & m n=25 \end{aligned}$