Question:

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