Question:

<div class="solution"><p><strong>Step 1:</strong> Factor: $(\sin^{-1}x-\cos^{-1}x)(\sin^{-1}x+\cos^{-1}x)>0$.</p><p><strong>Step 2:</strong> Second factor = $\pi/2>0$ always. So need $\sin^{-1}x>\cos^{-1}x$.</p><p><strong>Step 3:</strong> They intersect at $x=1/\sqrt{2}$. Since sin⁻^1 increasing and cos⁻^1 decreasing, sin⁻^1x>cos⁻^1x for $x>1/\sqrt{2}$.</p><p><strong>Answer: (1) $x\in(1/\sqrt{2},1]$</strong></p><div class="trap-box"><strong>Trap:</strong> After factorization, only the first factor decides the sign (second is always positive).<div class="key-concept"><strong>Key Concept:</strong> Difference of squares factorization + monotonicity of sin⁻^1 and cos⁻^1