Question:

<div class="solution"><p><strong>Step 1:</strong> $\cos\alpha=x\cdot\frac{y}{2}+\sqrt{1-x^2}\cdot\sqrt{1-y^2/4}\implies\sqrt{(1-x^2)(4-y^2)}=2\cos\alpha-xy$.</p><p><strong>Step 2:</strong> Square both sides: <span class="math-block">$$4-y^2-4x^2+x^2y^2=4\cos^2\alpha+x^2y^2-4xy\cos\alpha$$</p><p><strong>Step 3:</strong> Rearrange: $4x^2-4xy\cos\alpha+y^2=4-4\cos^2\alpha=4\sin^2\alpha$.</p><p><strong>Answer: (C) $4\sin^2\alpha$</strong></p><div class="trap-box"><strong>Trap:</strong> Arithmetic errors in expanding $(2\cos\alpha-xy)^2$.<div class="key-concept"><strong>Key Concept:</strong> cos⁻^1 subtraction formula + algebraic identity