Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>If <span style='font-family: Arial'>(sin A - sin C)/(cos C - cos A) = cot B</span>, then A, B and C are in</p>
<p>(a) AP</p>
<p>(b) GP</p>
<p>(c) HP</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Apply sum-to-product formulas to convert the trigonometric expression into a form revealing an arithmetic progression.
<p><strong>Step 1:</strong> Use sum-to-product formulas: <span style='font-family: Arial'>sin A - sin C = 2 cos((A+C)/2) sin((A-C)/2)</span></p><p><strong>Step 2:</strong> Similarly, <span style='font-family: Arial'>cos C - cos A = -2 sin((A+C)/2) sin((A-C)/2)</span></p><p><strong>Step 3:</strong> Substituting: <span style='font-family: Arial'>(sin A - sin C)/(cos C - cos A) = -cot((A+C)/2) = cot B</span></p><p><strong>Step 4:</strong> This gives <span style='font-family: Arial'>cot B = -cot((A+C)/2)</span>, which means <span style='font-family: Arial'>B = π - (A+C)/2</span> or <span style='font-family: Arial'>2B = A + C</span></p><p>∴ A, B, C are in AP. Answer is A.</p>
Correct Answer: A