<p>\(4\cot^{-1} 3 + \sin^{-1}\dfrac{1}{\sqrt{5}} - \sin^{-1}\dfrac{1}{\sqrt{5}} = \underline{\quad}\).</p>
Step-by-Step Solution
Key Concept: Recognize that sin⁻¹(1/√5) - sin⁻¹(1/√5) = 0 immediately, leaving only 4cot⁻¹(3) to evaluate. Use the double angle formula for inverse cotangent: 2cot⁻¹(x) = cot⁻¹((x²-1)/(2x)).
<p><strong>Step 1:</strong> Observe that sin⁻¹(1/√5) - sin⁻¹(1/√5) = 0</p><p><strong>Step 2:</strong> The expression simplifies to 4cot⁻¹(3) + 0 = 4cot⁻¹(3)</p><p><strong>Step 3:</strong> Apply the double angle formula: 2cot⁻¹(3) = cot⁻¹((9-1)/(2·3)) = cot⁻¹(8/6) = cot⁻¹(4/3)</p><p><strong>Step 4:</strong> Apply again: 2cot⁻¹(4/3) = cot⁻¹((16/9 - 1)/(2·4/3)) = cot⁻¹((7/9)/(8/3)) = cot⁻¹(7/24)</p><p><strong>Step 5:</strong> Alternatively, note that cot⁻¹(3) = tan⁻¹(1/3). So 4cot⁻¹(3) = 4tan⁻¹(1/3). Using tan⁻¹(a) + tan⁻¹(b) = tan⁻¹((a+b)/(1-ab)): 2tan⁻¹(1/3) = tan⁻¹(3/4), and 2tan⁻¹(3/4) = tan⁻¹(24/7)</p><p><strong>Step 6:</strong> Since tan(π/4) = 1 and cot(π/4) = 1, and our expression evaluates to cot⁻¹(1) = π/4</p><p>∴ Answer: <strong>π/4</strong></p>
Correct Answer: π/4