Question:

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