Question:

<div class="solution"><p><strong>Step 1:</strong> Let <span class="math-inline">\(\theta=\frac{1}{2}\cos^{-1}x\)</span> so <span class="math-inline">\(\cos 2\theta=x\)</span>.</p><p><strong>Step 2:</strong> <span class="math-block">\[\tan(\pi/4+\theta)+\tan(\pi/4-\theta)=\frac{1+\tan\theta}{1-\tan\theta}+\frac{1-\tan\theta}{1+\tan\theta}=\frac{2(1+\tan^2\theta)}{1-\tan^2\theta}=\frac{2}{\cos 2\theta}=\frac{2}{x}\]</span></p><p><strong>Answer: (C) <span class="math-inline">\(2/x\)</span></strong></p><div class="trap-box"><strong>Trap:</strong> Confusing with x or 2x — the cos 2θ = x appears in the denominator.</div><div class="key-concept"><strong>Key Concept:</strong> <span class="math-inline">\(\tan(\pi/4+\theta)+\tan(\pi/4-\theta)=2\sec 2\theta\)</span></div></div>