Question:

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