<p>\(2\sin^{-1}\sqrt{\dfrac{1-x}{2}} = \cos^{-1}(\underline{\quad})\).</p>
Step-by-Step Solution
Key Concept: Use the half-angle identity: if 2sin⁻¹(a) = cos⁻¹(b), then b = 1 - 2a². The argument √((1-x)/2) matches the half-angle form sin(θ/2) = √((1-cosθ)/2), connecting it to cosine of a double angle.
<p><strong>Step 1:</strong> Use the identity: 2sin⁻¹(a) = cos⁻¹(1 - 2a²) for a ∈ [0,1]</p><p><strong>Step 2:</strong> Let a = √((1-x)/2), then a² = (1-x)/2</p><p><strong>Step 3:</strong> Calculate 1 - 2a² = 1 - 2·(1-x)/2 = 1 - (1-x) = x</p><p><strong>Step 4:</strong> Therefore, 2sin⁻¹√((1-x)/2) = cos⁻¹(x)</p><p>∴ Answer: <strong>x</strong></p>
Correct Answer: x