If $m$ and $n$ respectively are the numbers of positive and negative value of $\theta$ in the interval $\left[-\pi ,\pi \right]$ that satisfy the equation $\cos 2\theta \cos \frac{\theta }{2}=\cos 3\theta \cos \frac{9\theta }{2}$, then $mn$ is equal to _____ .

We have,

$\cos 2\theta ·\cos \frac{\theta }{2}=\cos 3\theta ·\cos \frac{9\theta }{2}$

$\Rightarrow 2\cos 2\theta \cos \left(\frac{\theta }{2}\right)=2\cos \left(\frac{9\theta }{2}\right)\cos 3\theta$

$\Rightarrow \cos \left(\frac{5\theta }{2}\right)+\cos \left(\frac{3\theta }{2}\right)=\cos \left(\frac{15\theta }{2}\right)+\cos \left(\frac{3\theta }{2}\right)$

$\Rightarrow \cos \left(\frac{15\theta }{2}\right)=\cos \left(\frac{5\theta }{2}\right)$

$\Rightarrow \frac{15\theta }{2}=2\mathrm{k}\pi \pm \frac{5\theta }{2}; k\in Z$

$\Rightarrow 5\theta =2\mathrm{k}\pi$ or $10\theta =2\mathrm{k}\pi$

$\Rightarrow \theta =\frac{2k\pi }{5} \mathrm{or} \theta =\frac{k\pi }{5}$

$\therefore \theta =\left\{-\pi ,-\frac{4\pi }{5},-\frac{3\pi }{5},-\frac{2\pi }{5},-\frac{\pi }{5},0,\frac{\pi }{5},\frac{2\pi }{5},\frac{3\pi }{5},\frac{4\pi }{5},\pi \right\}$

So, $m=5,n=5$

$\therefore mn=25$.