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

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Detailed Solution

Given $\cos 2 θ . \cos \frac{θ}{2} = \cos 3 θ . \cos \frac{9 θ}{2}$
$\Rightarrow 2 \cos 2 θ . \cos \frac{θ}{2} = 2 \cos \frac{9 θ}{2} . \cos 3 θ$
By using property $2 \cos A \cos B = \cos (A + B) + \cos (A - B)$
$\Rightarrow \cos \frac{5 θ}{2} + \cos \frac{3 θ}{2} = \cos θ \frac{15 θ}{2} + \cos \frac{3 θ}{2}$
$\Rightarrow \cos \frac{15 θ}{2} = \cos \frac{5 θ}{2} \Rightarrow \frac{15 θ}{2} = 2 k π \pm \frac{5 θ}{2}$
$5 θ = 2 k π$ or $10 θ = 2 k π \Rightarrow θ = \frac{2 k π}{5} π = \frac{k π}{5}$
$∴ θ = {- π, \frac{- 4 π}{5}, \frac{- 3 π}{5}, \frac{- 2 π}{5}, \frac{- π}{5}, 0, \frac{π}{5}, \frac{2 π}{5}, \frac{3 π}{5}, \frac{4 π}{5}, π}$
$m = 5, n = 5 \Rightarrow m . n = 25$