<p>A value of \( x \) satisfying the equation \( \sin[\cot^{-1}(1+x)] = \cos[\tan^{-1}x] \) is</p>
Step-by-Step Solution
Key Concept: Convert both sides using the identity relationships in inverse trigonometric functions: if α = cot⁻¹(1+x) then sin α = 1/√((1+x)²+1), and if β = tan⁻¹(x) then cos β = 1/√(x²+1). Equate these expressions.
<p><strong>Step 1:</strong> Use the identity for sin(cot⁻¹(θ)). If cot⁻¹(1+x) = α, then cot α = 1+x, which gives sin α = 1/√((1+x)²+1).</p><p><strong>Step 2:</strong> Use the identity for cos(tan⁻¹(θ)). If tan⁻¹(x) = β, then tan β = x, which gives cos β = 1/√(x²+1).</p><p><strong>Step 3:</strong> Equate: 1/√((1+x)²+1) = 1/√(x²+1)</p><p><strong>Step 4:</strong> Square both sides: (1+x)²+1 = x²+1</p><p><strong>Step 5:</strong> Expand: 1 + 2x + x² + 1 = x² + 1</p><p><strong>Step 6:</strong> Simplify: 2x + 1 = 0, so x = -1/2</p><p><strong>Step 7:</strong> Verify: When x = -1/2, both sides evaluate to 2/√5, confirming the solution satisfies domain requirements.</p><p>∴ Answer: D</p>
Correct Answer: D