The number of solutions of the equation cos 2 cos 2 + cos 5 2 = 2 cos 3 5 2 in [ – 2 , 2 ] is: [2025]

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Solution

Q.
The number of solutions of the equation $\cos 2 θ \cos \frac{θ}{2} + \cos \frac{5 θ}{2} = 2 \cos^{3} \frac{5 θ}{2} in [- \frac{π}{2}, \frac{π}{2}]$ is: [2025]

1 5

2 9

3 7

4 6

Ans.
(3)

We have, $\cos 2 θ \cos \frac{θ}{2} + \cos \frac{5 θ}{2} = 2 \cos^{3} \frac{5 θ}{2}$

$\Rightarrow \frac{1}{2} (2 \cos 2 θ \cos \frac{θ}{2}) + \cos \frac{5 θ}{2} = \frac{1}{2} (\cos \frac{15 θ}{2} + 3 \cos \frac{5 θ}{2})$

$\Rightarrow \frac{1}{2} (\cos \frac{5 θ}{2} + \cos \frac{3 θ}{2}) = \cos \frac{5 θ}{2} \cos 5 θ$

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

$\Rightarrow \cos \frac{3 θ}{2} = \cos \frac{15 θ}{2} \Rightarrow \cos \frac{15 θ}{2} - \cos \frac{3 θ}{2} = 0$

$\Rightarrow 2 \sin 3 θ \sin \frac{9 θ}{2} = 0 \Rightarrow 3 θ = n π or \frac{9 θ}{2} = m π$

$\Rightarrow θ = \frac{n π}{3} or θ = \frac{2 m π}{9} for m, n \in Z$

$∴ θ = {- \frac{π}{3}, \frac{π}{3}, 0} and θ = {- \frac{4 π}{9}, \frac{- 2 π}{9}, 0, \frac{2 π}{9}, \frac{4 π}{9}}$

Required number of solutions = 7.

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