Question:

<div class="solution"><p><strong>Step 1:</strong> <span class="math-inline">\([a]=2\)</span> for <span class="math-inline">\(2\le a<3\)</span>.</p><p><strong>Step 2:</strong> <span class="math-inline">\(\cos^{-1}(\cos 2)=2\)</span> since <span class="math-inline">\(2\in[0,\pi]\)</span>.</p><p><strong>Step 3:</strong> <span class="math-inline">\(\text{cosec}^{-1}(\text{cosec }2)=\pi-2\)</span> since 2 is in 2nd quadrant and principal range requires <span class="math-inline">\(\pi-2\in[-\pi/2,\pi/2]\)</span>.</p><p><strong>Step 4:</strong> <span class="math-inline">\(\cot^{-1}(\cot 2)=2\)</span> since <span class="math-inline">\(2\in(0,\pi)\)</span>.</p><p><strong>Sum:</strong> <span class="math-inline">\(2+(\pi-2)+2=\pi+2\)</span></p><p><strong>Answer: (B) <span class="math-inline">\(\pi+2\)</span></strong></p><div class="trap-box"><strong>Trap:</strong> Assuming <span class="math-inline">\(f^{-1}(f(x))=x\)</span> always. It only holds within the principal range.</div><div class="key-concept"><strong>Key Concept:</strong> Check if angle is in principal range before applying <span class="math-inline">\(f^{-1}(f(\theta))=\theta\)</span></div></div>