If m and n respectively are the numbers of po | vedclass

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Question

$\cos 2 	heta \cdot \cos \frac{	heta}{2}=\cos 3 	heta \cdot \cos \frac{9 	heta}{2}$

$\Rightarrow 2 \cos 2 	heta \cdot \cos \frac{	heta}{2}=2

Vedclass · Accepted Answer

$25$

Vedclass · Answer

$\cos 2 	heta \cdot \cos \frac{	heta}{2}=\cos 3 	heta \cdot \cos \frac{9 	heta}{2}$

$\Rightarrow 2 \cos 2 	heta \cdot \cos \frac{	heta}{2}=2 \cos \frac{9 	heta}{2} \cdot \cos 3 	heta$

$\Rightarrow \cos \frac{5 	heta}{2}+\cos \frac{3 	heta}{2}=\cos \frac{15 	heta}{2}+\cos \frac{3 	heta}{2}$

$\Rightarrow \cos \frac{15 	heta}{2}=\cos \frac{5 	heta}{2}$

$\Rightarrow \frac{15 	heta}{2}=2 k \pi \pm \frac{5 	heta}{2}$

$5 	heta=2 k \pi 	ext { or } 10 	heta=2 k \pi$

$	heta=\frac{2 k \pi}{5}$

$	herefore 	heta=\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ight\}$

$m =5, n =5$

$	herefore m n=25$

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

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