If m and n respectively are the numbers of positive and negative values of $\theta$ in the interval $[-\pi, \pi]$ that satisfy the equation $\cos 2\theta \cos\frac{\theta}{2} = \cos 3\theta \cos\frac{9\theta}{2}$, then $mn$ is equal to _____.
Step-by-Step Solution
Key Concept: Convert the product-to-sum identity on both sides, reduce to a single cosine equation, list solutions in $[-\pi, \pi]$.
$\cos\frac{15\theta}{2} = \cos\frac{5\theta}{2} \Rightarrow \frac{15\theta}{2} = \pm\frac{5\theta}{2} + 2k\pi$. Case 1: $5\theta = 2k\pi$, Case 2: $10\theta = 2k\pi$, i.e., $\theta = k\pi/5$. Values in $[-\pi,\pi]$: $\{-\pi, -4\pi/5, -3\pi/5, -2\pi/5, -\pi/5, 0, \pi/5, 2\pi/5, 3\pi/5, 4\pi/5, \pi\}$. $m=5$ (positive), $n=5$ (negative). $mn=25$. Answer: 25
Correct Answer: 25