Trigonometry & Inverse Trigonometry
Trigonometric Identities
Grade 11
Question:
<p><strong>Assertion (A):</strong> The value of <span>tan 3α · cot α</span> cannot lie between 3 and 1/3.</p><p><strong>Reason (R):</strong> In a triangle ABC, the maximum value of <span>sin(A/2) sin(B/2) sin(C/2)</span> is <span>1/8</span>.</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: Both statements are independently true; the assertion about tan 3α relies on algebraic manipulation, while the reason about triangle inequalities uses calculus/AM-GM.
<p><strong>Analysis of Assertion:</strong> Using <span>tan 3α = (3 tan α - tan³ α)/(1 - 3tan²α)</span>, we get <span>tan 3α · cot α = (3 - tan²α)/(1 - 3tan²α)</span>.</p><p>Let <span>t = tan²α</span>. Then <span>f(t) = (3 - t)/(1 - 3t)</span>. The function excludes values in <span>(1/3, 3)</span>.</p><p><strong>Analysis of Reason:</strong> By AM-GM inequality applied to the constraint <span>A + B + C = π</span>, the maximum value of <span>sin(A/2) sin(B/2) sin(C/2)</span> is indeed <span>1/8</span>, achieved when <span>A = B = C = π/3</span>.</p><p>Both A and R are true, but R does not explain A (they concern different topics).</p><p>∴ Answer is (B).</p>
Correct Answer: B