Question:

<div class="solution"><p><strong>Step 1:</strong> Let $x=\tan\theta$, $x>1\implies\theta\in(\pi/4,\pi/2)$.</p><p><strong>Step 2:</strong> $\beta=\sin^{-1}(\cos 2\theta)=\sin^{-1}(\sin(\pi/2-2\theta))$. Since $\pi/2-2\theta\in(-\pi/2,0)$: $\beta=\pi/2-2\theta$.</p><p><strong>Step 3:</strong> $\frac{1+x}{1-x}=\frac{1+\tan\theta}{1-\tan\theta}=\tan(\pi/4+\theta)$. Since $\pi/4+\theta>\pi/2$: $\alpha=2(\pi/4+\theta-\pi)=2\theta-3\pi/2$.</p><p><strong>Step 4:</strong> $\alpha+\beta=(2\theta-3\pi/2)+(\pi/2-2\theta)=-\pi$.</p><p><strong>Answer: (A) $-\pi$</strong></p><div class="trap-box"><strong>Trap:</strong> Not subtracting \pi when tan⁻^1 argument exceeds the principal range.<div class="key-concept"><strong>Key Concept:</strong> Branch tracking is mandatory when argument of tan⁻^1 moves outside principal interval