If m and n respectively are the numbers of positive and negative...

We have,

$\cos 2\theta \cdot cos\dfrac{\theta}{2} = \cos 3\theta \cdot cos\dfrac{9\theta}{2}$

$⇒ 2\cos 2\theta cos\left(\dfrac{\theta}{2}\right) = 2\cos \left(\dfrac{9\theta}{2}\right)\cos 3\theta$

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

$⇒ \cos \left(\dfrac{15\theta}{2}\right) = \cos \left(\dfrac{5\theta}{2}\right)$

$⇒ \dfrac{15\theta}{2} = 2k\pi \pm \dfrac{5\theta}{2}; k \in Z$

$⇒ 5\theta = 2k\pi$ or $10\theta = 2k\pi$

$⇒ \theta = \dfrac{2k\pi}{5}or\theta = \dfrac{k\pi}{5}$

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

So, $m = 5, n = 5$

$∴ mn = 25$.