Question:

<div class="solution"><p><strong>Key Idea:</strong> Write x = n\pi+t with 0\let<\pi. Then {x/\pi}=t/\pi.</p><p><strong>Step 1:</strong> $f(x)=2(n\pi+t)-t/\pi=2n\pi+(2-1/\pi)t$</p><p><strong>Step 2:</strong> $(g\circ f)(x)=\cos((2-1/\pi)t)$ -- depends only on t = x mod \pi.</p><p><strong>Step 3:</strong> So period is \pi. Verified: \pi/2 fails by direct test.</p><p><strong>Answer: \pi</strong></p><div class="trap-box"><strong>Trap:</strong> The 2n\pi disappears inside cosine, but you must decompose x into n\pi+t to see this.<div class="key-concept"><strong>Key Concept:</strong> {x/\pi} inside trig composition -- decompose by \pi-quotient and remainder