<p>\(\sin^{-1}\sqrt{\dfrac{x}{x+y}} = \tan^{-1}(\underline{\quad})\).</p>
Step-by-Step Solution
Key Concept: Use the right triangle relationship where if sin⁻¹(a) = θ, then sin(θ) = a. Construct a right triangle with opposite = √x and hypotenuse = √(x+y), then find tan(θ) using the Pythagorean theorem to get the adjacent side.
<p><strong>Step 1:</strong> Let sin⁻¹√(x/(x+y)) = θ, so sin(θ) = √(x/(x+y))</p><p><strong>Step 2:</strong> In a right triangle, sin(θ) = opposite/hypotenuse. Set opposite = √x and hypotenuse = √(x+y)</p><p><strong>Step 3:</strong> Find adjacent using Pythagorean theorem: adjacent² = (√(x+y))² - (√x)² = (x+y) - x = y, so adjacent = √y</p><p><strong>Step 4:</strong> Therefore, tan(θ) = opposite/adjacent = √x/√y = √(x/y)</p><p><strong>Step 5:</strong> Since sin⁻¹√(x/(x+y)) = θ and tan(θ) = √(x/y), we have sin⁻¹√(x/(x+y)) = tan⁻¹(√(x/y))</p><p>∴ Answer: <strong>√(x/y)</strong></p>
Correct Answer: √(x/y)