If m and n respectively are the numbers of positive and negative values of in the interval…

Question

If $m$ and $n$ respectively are the numbers of positive and negative values of $	heta$ in the interval $[-\pi,\pi]$ that satisfy the equation $\cos2	heta\cos\dfrac	heta2 = \cos3	heta\cos\dfrac{9	heta}2$, then $mn$ is equal to \_\_\_\_\_\_\_\_\_\_\_

Answer: 25

Accepted Answer

$$\cos2\theta\cos\tfrac\theta2 = \cos3\theta\cos\tfrac{9\theta}2 \implies 2\cos2\theta\cos\tfrac\theta2 = 2\cos\tfrac{9\theta}2\cos3\theta$$

Using product-to-sum:

$$\cos\tfrac{5\theta}2+\cos\tfrac{3\theta}2 = \cos\tfrac{15\theta}2+\cos\tfrac{3\theta}2 \implies \cos\tfrac{15\theta}2 = \cos\tfrac{5\theta}2$$

$$\frac{15\theta}2 = 2k\pi \pm \frac{5\theta}2 \implies 5\theta = 2k\pi \ \text{ or } \ 10\theta = 2k\pi$$

Both give $\theta = \dfrac{2k\pi}5$ (the second is a subset of the first pattern once all integers $k$ are allowed). Within $[-\pi,\pi]$:

$$\theta = -\pi,\ -\tfrac{4\pi}5,\ -\tfrac{3\pi}5,\ -\tfrac{2\pi}5,\ -\tfrac\pi5,\ 0,\ \tfrac\pi5,\ \tfrac{2\pi}5,\ \tfrac{3\pi}5,\ \tfrac{4\pi}5,\ \pi$$

There are $m=5$ positive values and $n=5$ negative values (excluding $0,\pm\pi$ from the count as appropriate), so $mn = 25$.

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