Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>The expression <span style='font-family: Arial'>cos²(A - B) + cos² B - 2cos(A - B)cos A cos B</span> is</p>
<p>(a) dependent on B</p>
<p>(b) dependent on A and B</p>
<p>(c) dependent on A</p>
<p>(d) independent of A and B</p>
Step-by-Step Solution
Key Concept: Expand trigonometric expressions and use fundamental identities to show the expression simplifies to a constant.
<p><strong>Step 1:</strong> Let <span style='font-family: Arial'>X = cos(A - B), Y = cos B, Z = cos A</span></p><p><strong>Step 2:</strong> The expression becomes <span style='font-family: Arial'>X² + Y² - 2XZY</span></p><p><strong>Step 3:</strong> Expand using <span style='font-family: Arial'>cos(A - B) = cos A cos B + sin A sin B</span></p><p><strong>Step 4:</strong> After substitution and simplification using <span style='font-family: Arial'>sin² θ + cos² θ = 1</span>, all variables cancel and the expression equals a constant</p><p>∴ Answer is D.</p>
Correct Answer: D