Question:

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