Trigonometry & Inverse Trigonometry
Grade 12
Question:
<p>If \(\cos^{-1}x-\dfrac{y}{2}=\alpha\), then \(4x^2-4xy\cos\alpha+y^2=\)</p>
4x^2-4xy cdot cosα+y^2
4x^2-4xy cdot cosα-y^2
4x^2+4xy cdot cosα+y^2
4x^2+4xy cdot cosα-y^2
Step-by-Step Solution
<div class="solution"><p>Same identity as O1-Q15. Rearrange: $\cos^{-1}x=\alpha+y/2\implies x=\cos(\alpha+y/2)$. Substitute into the quadratic form and simplify using cos addition: result is $4\sin^2\alpha$.</p><div class="key-concept"><strong>Key Concept:</strong> Standard algebraic identity for nested cosines
Correct Answer: 4sin²α