If \( 2 \sec 4\beta = \tan 2\alpha + \cot 2\alpha \), then which one of the following is a possible value of \( (\alpha + \beta) \) ?
- A. \frac{\pi}{2}
- B. \frac{\pi}{4}
- C. \frac{\pi}{6}
- D. \frac{\pi}{8} ✓
Correct Answer: D. \frac{\pi}{8}
Explanation
\tan 2\alpha + \cot 2\alpha = \frac{2}{\sin 4\alpha}. So, 2\sec 4\beta = 2\operatorname{cosec} 4\alpha. This means \sec 4\beta = \operatorname{cosec} 4\alpha \implies \cos 4\beta = \sin 4\alpha. This holds when 4\alpha + 4\beta = \frac{\pi}{2}, so \alpha + \beta = \frac{\pi}{8}.