Question

If $m$ and $n$ respectively are the numbers of positive and negative values of $\theta$ in the interval $[-\pi, \pi]$ that satisfy the equation $\cos 2 \theta \cos \frac{\theta}{2}=\cos 3 \theta \cos \frac{99}{2}$, then $m n$ is equal to__

Answer 1

$ \cos 2 \theta \cdot \cos \frac{\theta}{2}=\cos 3 \theta \cdot \cos \frac{9 \theta}{2} $
$ \Rightarrow 2 \cos 2 \theta \cdot \cos \frac{\theta}{2}=2 \cos \frac{9 \theta}{2} \cdot \cos 3 \theta$
$ \Rightarrow \cos \frac{5 \theta}{2}+\cos \frac{3 \theta}{2}=\cos \frac{15 \theta}{2}+\cos \frac{3 \theta}{2} $
$ \Rightarrow \cos \frac{15 \theta}{2}=\cos \frac{5 \theta}{2} $
$\Rightarrow \frac{15 \theta}{2}=2 k \pi \pm \frac{5 \theta}{2}$
$ 5 \theta=2 k \pi \text { or } 10 \theta=2 k \pi$
$\theta=\frac{2 k \pi}{5} \theta=\frac{ k \pi}{5}$
$\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\} $
$m =5, n =5 $
$ \therefore m \cdot n =25$