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