Question:

<div class="solution"><p><strong>Key Idea:</strong> Let <span class="math-inline">\(t=x^2+1/x^2\ge 2\)</span> (AM-GM).</p><p><strong>Step 1:</strong> Expression is <span class="math-inline">\(\tan^{-1}(1-t)+\sin^{-1}(t-1)\)</span>.</p><p><strong>Step 2:</strong> <span class="math-inline">\(\sin^{-1}(t-1)\)</span> requires <span class="math-inline">\(t-1\in[-1,1]\implies t\in[0,2]\)</span>. Combined with <span class="math-inline">\(t\ge 2\)</span>: <span class="math-inline">\(t=2\)</span> only.</p><p><strong>Step 3:</strong> At <span class="math-inline">\(t=2\)</span>: <span class="math-inline">\(\tan^{-1}(-1)+\sin^{-1}(1)=-\pi/4+\pi/2=\pi/4\)</span></p><p><strong>Answer: (B) <span class="math-inline">\(\pi/4\)</span></strong></p><div class="trap-box"><strong>Trap:</strong> Looking for a trig identity before checking domain — the domain forces t=2.</div><div class="key-concept"><strong>Key Concept:</strong> Check domain constraints first; often they force a unique value</div></div>