Question

If $m$ and $n$ respectively are the numbers of positive and negative values of $\theta$ in the interval $[-\pi, \pi]$ that satisfy the equation $\cos 2 \theta \cos \frac{\theta}{2}=\cos 3 \theta \cos \frac{99}{2}$, then $m n$ is equal to__

Answer 1

The correct answer is 25.
cos2θ⋅cos2θ​=cos3θ⋅cos29θ​
⇒2cos2θ⋅cos2θ​=2cos29θ​⋅cos3θ
⇒cos25θ​+cos23θ​=cos215θ​+cos23θ​
⇒cos215θ​=cos25θ​
⇒215θ​=2kπ±25θ​
5θ=2kπ or 10θ=2kπ
θ=52kπ​θ=5kπ​
∴θ={−π,5−4π​,5−3π​,5−2π​,5−π​,0,5π​,52π​,53π​,54π​,π}
m=5,n=5
∴m⋅n=25