<p>The functions \(f(x) = \cos^{-1}\sqrt{1-x^2}\) and \(g(x) = \sin^{-1}x\) are identical for \(x\) belonging to</p>
Step-by-Step Solution
Key Concept: Two functions are identical if they have the same domain AND produce identical outputs for all x in that domain. Use the identity cos⁻¹(√(1-x²)) = sin⁻¹|x| and carefully track domain restrictions for inverse functions.
<p><strong>Step 1:</strong> Find the domain of f(x) = cos⁻¹√(1-x²). We need 1-x² ≥ 0, so x ∈ [-1,1].</p><p><strong>Step 2:</strong> Use the identity: cos⁻¹(√(1-x²)) = sin⁻¹|x|. This is because if cos(θ) = √(1-x²) where θ ∈ [0,π], then sin(θ) = √(1-cos²(θ)) = √(1-(1-x²)) = |x|, giving θ = sin⁻¹|x|.</p><p><strong>Step 3:</strong> Compare f(x) = sin⁻¹|x| with g(x) = sin⁻¹(x).</p><p>• For x ≥ 0: sin⁻¹|x| = sin⁻¹(x), so f(x) = g(x) ✓</p><p>• For x < 0: sin⁻¹|x| = sin⁻¹(-x) ≠ sin⁻¹(x), so f(x) ≠ g(x) ✗</p><p><strong>Step 4:</strong> The functions are identical only when x ≥ 0. Combined with domain x ∈ [-1,1], we get x ∈ [0,1].</p><p>∴ Answer: B (x ∈ [0,1])</p>
Correct Answer: B