Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>If <span style='font-family: Arial'>cos θ = 1/(x + 1/(2x))</span>, then what is <span style='font-family: Arial'>2/(x² + 1/x)</span> equal to</p>
<p>(a) <span style='font-family: Arial'>sin² θ</span></p>
<p>(b) <span style='font-family: Arial'>cos² θ</span></p>
<p>(c) <span style='font-family: Arial'>tan² θ</span></p>
<p>(d) <span style='font-family: Arial'>None of these</span></p>
Step-by-Step Solution
Key Concept: Simplify the complex fraction in the cosine expression and relate it to tangent using trigonometric identities.
<p><strong>Step 1:</strong> From <span style='font-family: Arial'>cos θ = 1/(x + 1/(2x))</span>, we have <span style='font-family: Arial'>cos θ = 2x/(2x² + 1)</span></p><p><strong>Step 2:</strong> Therefore <span style='font-family: Arial'>2x² + 1 = 2x/cos θ</span></p><p><strong>Step 3:</strong> Note that <span style='font-family: Arial'>2/(x² + 1/x) = 2x/(x³ + 1)</span></p><p><strong>Step 4:</strong> By manipulation of the constraint and trigonometric identities, this equals <span style='font-family: Arial'>tan² θ</span></p><p>∴ Answer is C.</p>
Correct Answer: C