Question:

<div class="solution"><p><strong>Step 1:</strong> Need \(1-x^2>0\implies x\in(-1,1)\).</p><p><strong>Step 2:</strong> \(1-x^2\in(0,1]\), so \(\log_e(1-x^2)\in(-\infty,0]\).</p><p><strong>Step 3:</strong> \(\cot^{-1}\) is decreasing: \(\cot^{-1}(0)=\pi/2\) and \(\lim_{u\to-\infty}\cot^{-1}(u)=\pi\).</p><p><strong>Answer: (C) \([\pi/2,\pi)\)</strong></p><div class="trap-box"><strong>Trap:</strong> Forgetting that \(\cot^{-1}\) is <em>decreasing</em> -- the range flips direction.<div class="key-concept"><strong>Key Concept:</strong> Decreasing functions reverse inequality direction in range calculations