Which one of the following is a value of \theta, if \theta satisfies the equation \tan 2\theta \tan 4\theta-1=0, 0 \lt \theta \lt \frac{\pi}{2}?
- A. \frac{\pi}{12} ✓
- B. \frac{\pi}{15}
- C. \frac{\pi}{6}
- D. \frac{\pi}{5}
Correct Answer: A. \frac{\pi}{12}
Explanation
The equation is \tan 2\theta \tan 4\theta = 1, which implies \tan 4\theta = \frac{1}{\tan 2\theta} = \cot 2\theta. We know \cot 2\theta = \tan(\frac{\pi}{2} - 2\theta). Therefore, 4\theta = \frac{\pi}{2} - 2\theta \implies 6\theta = \frac{\pi}{2} \implies \theta = \frac{\pi}{12}.
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