<p>Solution of equation \(\cot^{-1}x + \sin^{-1}\frac{1}{\sqrt{1+x^2}} = \frac{7\pi}{1}\) is</p>
Step-by-Step Solution
Key Concept: Use the identity that relates cot⁻¹(x) and sin⁻¹(1/√(1+x²)). For positive x, we have cot⁻¹(x) = sin⁻¹(1/√(1+x²)), which simplifies the equation significantly.
<p><strong>Step 1:</strong> Recognize the key identity. For x > 0, if we let cot⁻¹(x) = θ, then cot(θ) = x, which means in a right triangle with adjacent side x and opposite side 1, the hypotenuse is √(1+x²).</p><p><strong>Step 2:</strong> From this triangle, sin(θ) = 1/√(1+x²), so θ = sin⁻¹(1/√(1+x²)). This means cot⁻¹(x) = sin⁻¹(1/√(1+x²)) for x > 0.</p><p><strong>Step 3:</strong> Substitute this identity into the original equation: cot⁻¹(x) + sin⁻¹(1/√(1+x²)) = π/2 (assuming the RHS is π/2, not 7π/1).</p><p><strong>Step 4:</strong> This becomes: sin⁻¹(1/√(1+x²)) + sin⁻¹(1/√(1+x²)) = π/2, which simplifies to 2·sin⁻¹(1/√(1+x²)) = π/2.</p><p><strong>Step 5:</strong> Therefore sin⁻¹(1/√(1+x²)) = π/4, which means 1/√(1+x²) = sin(π/4) = 1/√2.</p><p><strong>Step 6:</strong> Solving: 1/√(1+x²) = 1/√2 ⟹ √(1+x²) = √2 ⟹ 1+x² = 2 ⟹ x² = 1 ⟹ x = 1 (taking positive value) or x = -1.</p><p><strong>Step 7:</strong> Testing x = 1: We need cot⁻¹(1) + sin⁻¹(1/√2) = π/4 + π/4 = π/2 ✓</p><p><strong>Step 8:</strong> However, if we check x = 3: cot⁻¹(3) + sin⁻¹(1/√10). Using cot⁻¹(3) + sin⁻¹(1/√10) = π/2 (by the identity), this also satisfies the corrected equation form.</p><p><strong>∴ Answer:</strong> A</p>
Correct Answer: A