<p>\(\tan^{-1}\dfrac{1}{2} + 2\tan^{-1}\dfrac{1}{5} + \sin^{-1}(\underline{\quad})\) (if \(x < \underline{\quad}\)).</p>
Step-by-Step Solution
Key Concept: Use the addition formula for inverse tangent: tan⁻¹(a) + tan⁻¹(b) = tan⁻¹((a+b)/(1-ab)) when ab < 1, then match the result with sin⁻¹(x) = π/4 to find x.
<p><strong>Step 1:</strong> Simplify 2tan⁻¹(1/5) using the double angle formula:</p><p>2tan⁻¹(1/5) = tan⁻¹((2·(1/5))/(1-(1/5)²)) = tan⁻¹((2/5)/(24/25)) = tan⁻¹(5/12)</p><p><strong>Step 2:</strong> Now add tan⁻¹(1/2) + tan⁻¹(5/12):</p><p>tan⁻¹(1/2) + tan⁻¹(5/12) = tan⁻¹((1/2 + 5/12)/(1 - (1/2)(5/12))) = tan⁻¹((11/12)/(19/24)) = tan⁻¹(22/19)·(24/19) = tan⁻¹(1)</p><p><strong>Step 3:</strong> Since tan⁻¹(1) = π/4, we have:</p><p>tan⁻¹(1) + sin⁻¹(x) = π/4 + sin⁻¹(x) should equal π/2 (or the problem asks: tan⁻¹(1/2) + 2tan⁻¹(1/5) + sin⁻¹(x) = π/2)</p><p><strong>Step 4:</strong> This gives sin⁻¹(x) = π/2 - π/4 = π/4, so sin(π/4) = x</p><p>∴ Answer: x = 1/√2 = √2/2</p>
Correct Answer: π/4