<p>If \(x + \frac{1}{x} = 2\), the principal value of \(\sin^{-1}x\) is</p>
Step-by-Step Solution
Key Concept: Start by solving the equation x + 1/x = 2 to find the value of x, then determine which value lies in the range of sin⁻¹ and equals the sine of the argument.
<p><strong>Step 1:</strong> Solve x + 1/x = 2</p><p>Multiply both sides by x: x² + 1 = 2x</p><p>Rearrange: x² - 2x + 1 = 0</p><p><strong>Step 2:</strong> Factor the quadratic</p><p>(x - 1)² = 0</p><p>Therefore: x = 1 (double root)</p><p><strong>Step 3:</strong> Verify x is in the domain of sin⁻¹</p><p>The domain of sin⁻¹ is [-1, 1]. Since x = 1, it is valid.</p><p><strong>Step 4:</strong> Find sin⁻¹(1)</p><p>We need the value θ such that sin(θ) = 1 and θ ∈ [-π/2, π/2]</p><p>The angle in [-π/2, π/2] where sin(θ) = 1 is θ = π/2</p><p><strong>Step 5:</strong> Verification</p><p>sin(π/2) = 1 ✓, and π/2 is in the range [-π/2, π/2] ✓</p><p><strong>∴ Answer: D</strong></p>
Correct Answer: D