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 ______.

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Trigonometric Equations

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 mn is equal to ______.

Answer: 25

Doubling and applying

2 cos A cos B = cos (A + B) + cos (A - B)

to both sides turns the equation into

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

, which reduces to

cos \frac{15 θ}{2} = cos \frac{5 θ}{2}

. This needs

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

, giving

θ = \frac{k π}{5}

(the family

θ = \frac{2 k π}{5}

is already contained in it). Within

[- π, π]

these are

θ = \frac{k π}{5}

for

k = - 5, \dots, 5

, with five positive and five negative values, so

m = 5

n = 5

and

mn = 25

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Integer answer