Trigonometry & Inverse Trigonometry
Optimization
Grade 11
Question:
<p><strong>Assertion (A):</strong> The minimum value of <span>a² tan²θ + b² cot²θ</span> is <span>2ab</span>.</p><p><strong>Reason (R):</strong> For positive real numbers AM ≥ GM.</p>
<p>(A) Both A and R are true and R is the correct explanation of A.</p>
<p>(B) Both A and R are true but R is not the correct explanation of A.</p>
<p>(C) A is true but R is false.</p>
<p>(D) A is false but R is true.</p>
Step-by-Step Solution
Key Concept: The AM-GM inequality applied to the two terms directly yields the minimum value.
<p><strong>Proof:</strong> By AM-GM inequality:</p><p><span>a² tan²θ + b² cot²θ ≥ 2√(a² tan²θ · b² cot²θ) = 2√(a²b² tan²θ cot²θ) = 2√(a²b²) = 2ab</span></p><p>Equality holds when <span>a² tan²θ = b² cot²θ</span>, i.e., when <span>tan θ = b/a</span>.</p><p>Thus the minimum value is <span>2ab</span>, and the AM-GM inequality is the direct reason.</p><p>∴ Answer is (A).</p>
Correct Answer: A