Question:

<div class="solution"><p><strong>Key Idea:</strong> Domain of <span class="math-inline">\(\text{cosec}^{-1}(u)\)</span> requires <span class="math-inline">\(|u|\ge 1\)</span>.</p><p><strong>Step 1:</strong> Need <span class="math-inline">\(|\cos x|\ge 1\)</span>. Since <span class="math-inline">\(\cos x\in[-1,1]\)</span>, we need <span class="math-inline">\(|\cos x|=1\)</span>.</p><p><strong>Step 2:</strong> <span class="math-inline">\(\cos x=\pm 1\)</span> iff <span class="math-inline">\(x=n\pi\)</span>.</p><p><strong>Answer: (D) x is a multiple of π</strong></p><div class="trap-box"><strong>Trap:</strong> Confusing with odd multiples of π/2 (where cos x = 0, not ±1).</div><div class="key-concept"><strong>Key Concept:</strong> cosec⁻¹ and sec⁻¹ need |argument| ≥ 1 — only attained at extremes of cos/sin</div></div>