Question:

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