If m and n respectively are the numbers of positive and negative values of in the interval that satisfy the equation cos 2 cos 2 = cos 3 cos 9 2 , then m n is equal to _______. [2023]

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Solution

Q.
If $m$ and $n$ respectively are the numbers of positive and negative values 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 $m n$ is equal to _______. [2023]

Ans.
(25)

$We have, \cos 2 θ \cdot \cos \frac{θ}{2} = \cos 3 θ \cdot \cos \frac{9 θ}{2}$

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

$\Rightarrow \cos \frac{5 θ}{2} + \cos \frac{3 θ}{2} = \cos \frac{15 θ}{2} + \cos \frac{3 θ}{2}$

$\Rightarrow \cos \frac{15 θ}{2} = \cos \frac{5 θ}{2} \Rightarrow \frac{15 θ}{2} = 2 k π \pm \frac{5 θ}{2}$

$\Rightarrow 5 θ = 2 k π or 10 θ = 2 k π \Rightarrow θ = \frac{2 k π}{5} or θ = \frac{k π}{5}$

$\Rightarrow θ = \frac{k π}{5}$

$Also, - π \leq \frac{k π}{5} \leq π \Rightarrow - 5 \leq k \leq 5$ $∴$ $m = 5, n = 5$

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