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જો $m$ અને $n$ એ સમીકરણ $\cos 2 \theta \cos \frac{\theta}{2}=\cos 3 \theta \cos \frac{9 \theta}{2}$ નું સમાધાન કરતી અંતરાલ $[-\pi, \pi]$ માં ની $\theta$ ની અનુક્રમે ધન અને ઋણ કિંમતો હોય, તો $m n=...........$
$25$
$24$
$23$
$22$
Solution
$\cos 2 \theta \cdot \cos \frac{\theta}{2}=\cos 3 \theta \cdot \cos \frac{9 \theta}{2}$
$\Rightarrow 2 \cos 2 \theta \cdot \cos \frac{\theta}{2}=2 \cos \frac{9 \theta}{2} \cdot \cos 3 \theta$
$\Rightarrow \cos \frac{5 \theta}{2}+\cos \frac{3 \theta}{2}=\cos \frac{15 \theta}{2}+\cos \frac{3 \theta}{2}$
$\Rightarrow \cos \frac{15 \theta}{2}=\cos \frac{5 \theta}{2}$
$\Rightarrow \frac{15 \theta}{2}=2 k \pi \pm \frac{5 \theta}{2}$
$5 \theta=2 k \pi \text { or } 10 \theta=2 k \pi$
$\theta=\frac{2 k \pi}{5}$
$\therefore \theta=\left\{-\pi, \frac{-4 \pi}{5}, \frac{-3 \pi}{5}, \frac{-2 \pi}{5}, \frac{-\pi}{5}, 0, \frac{\pi}{5}, \frac{2 \pi}{5}, \frac{3 \pi}{5}, \frac{4 \pi}{5}, \pi\right\}$
$m =5, n =5$
$\therefore m n=25$
