<p>\(\sin[\cot^{-1}\{\tan(\cos^{-1} x)\}]\) is equal to</p>
Step-by-Step Solution
Key Concept: Work from innermost function outward: use cos⁻¹(x) = θ to get cos(θ) = x, then find tan(θ) using the right triangle, apply cot⁻¹ to get another angle, and finally find sin of that angle using another right triangle.
<p><strong>Step 1:</strong> Let cos⁻¹(x) = α, so cos(α) = x where α ∈ [0, π]</p><p><strong>Step 2:</strong> Find tan(α): From cos(α) = x, we get sin(α) = √(1 - x²) (positive since α ∈ [0, π])</p><p>Therefore, tan(α) = sin(α)/cos(α) = √(1 - x²)/x</p><p><strong>Step 3:</strong> Let cot⁻¹{tan(α)} = β, so cot(β) = √(1 - x²)/x where β ∈ (0, π)</p><p><strong>Step 4:</strong> Find sin(β): If cot(β) = √(1 - x²)/x, then tan(β) = x/√(1 - x²)</p><p>Using the identity sin²(β) + cos²(β) = 1 and tan(β) = sin(β)/cos(β):</p><p>sin(β) = tan(β)/√(1 + tan²(β)) = [x/√(1 - x²)]/√[1 + x²/(1 - x²)]</p><p>= [x/√(1 - x²)]/√[(1 - x² + x²)/(1 - x²)] = [x/√(1 - x²)] · √(1 - x²)/1</p><p><strong>Step 5:</strong> sin(β) = x</p><p>∴ Answer: <strong>C</strong> (which is <strong>x</strong>)</p>
Correct Answer: C