Question:

<div class="solution"><p><strong>Step 1:</strong> Let $\theta=\cos^{-1}x\in[0,\pi]$. Equation: $\cos^{-1}(\cos 2\theta)=2\pi-2\theta$.</p><p><strong>Step 2:</strong> $\cos^{-1}(\cos\alpha)=2\pi-\alpha$ is valid for $\alpha\in[\pi,2\pi]$.</p><p><strong>Step 3:</strong> Need $2\theta\in[\pi,2\pi]\implies\theta\in[\pi/2,\pi]\implies\cos^{-1}x\in[\pi/2,\pi]\implies x\in[-1,0]$.</p><p><strong>Answer: (A) $x\in[-1,0]$</strong></p><div class="trap-box"><strong>Trap:</strong> Standard property: $\cos^{-1}(2x^2-1)=2\cos^{-1}x$ for $x\in[0,1]$ and $2\pi-2\cos^{-1}x$ for $x\in[-1,0]$.<div class="key-concept"><strong>Key Concept:</strong> Branch-tracking in double-angle cos⁻^1 identities