Question:

<div class="solution"><p><strong>Step 1:</strong> Domain: need \(|\tan x|\le 1\) AND \(|\cot x|\le 1\), i.e., \(|\tan x|\ge 1\). So \(|\tan x|=1\implies x=n\pi\pm\pi/4\). (B) ✓</p><p><strong>Step 2:</strong> At \(x=n\pi+\pi/4\): \(f=\sin^{-1}(1)+\cos^{-1}(1)=\pi/2\). At \(x=n\pi-\pi/4\): \(f=\sin^{-1}(-1)+\cos^{-1}(-1)=-\pi/2+\pi=\pi/2\). (A) ✓</p><p><strong>Step 3:</strong> Points spaced by \(\pi/2\) \to period \(\pi/2\). (C) ✓. Multiple x \to same y \to many-one. (D) ✓</p><p><strong>Answer: All four (A),(B),(C),(D)</strong></p><div class="trap-box"><strong>Trap:</strong> Assuming \(\sin^{-1}u+\cos^{-1}u=\pi/2\) applies everywhere -- here the arguments are equal only at specific points.<div class="key-concept"><strong>Key Concept:</strong> |t|\le1 AND |1/t|\le1 forces |t|=1