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 $mn$ is equal to__

Answer 1

Answer: 25

$\cos 2\theta \cdot \cos \frac{\theta }{2}=\cos 3\theta \cdot \cos \frac{9\theta }{2}$
$\Rightarrow 2\cos 2\theta \cdot \cos \frac{\theta }{2}=2\cos \frac{9\theta }{2}\cdot \cos 3\theta$
$\Rightarrow \cos \frac{5\theta }{2}+\cos \frac{3\theta }{2}=\cos \frac{15\theta }{2}+\cos \frac{3\theta }{2}$
$\Rightarrow \cos \frac{15\theta }{2}=\cos \frac{5\theta }{2}$
$\Rightarrow \frac{15\theta }{2}=2k\pi \pm \frac{5\theta }{2}$
$5\theta =2k\pi \text{ or }10\theta =2k\pi$
$\theta =\frac{2k\pi }{5}\theta =\frac{k\pi }{5}$
$\therefore \theta =\left\{-\pi ,\frac{-4\pi }{5},\frac{-3\pi }{5},\frac{-2\pi }{5},\frac{-\pi }{5},0,\frac{\pi }{5},\frac{2\pi }{5},\frac{3\pi }{5},\frac{4\pi }{5},\pi \right\}$
$m=5,n=5$
$\therefore m\cdot n=25$