Trigonometry & Inverse Trigonometry
Trigonometric identities
Grade 11
Question:
<p>If \(\cot y = \frac{\sin x - \sin z}{\cos z - \cos x}\) then which of the following is possible?</p>
<p>(a) \(x\), \(y\), \(z\) are in AP</p>
<p>(b) \(-x\), \(y\), \(z\) are in AP</p>
<p>(c) \(x\), \(y\), \(z\) are in HP</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: Use sum-to-product formulas to transform the numerator and denominator, recognizing that the expression simplifies to cot of a specific angle related to x and z.
<p><strong>Step 1:</strong> Apply sum-to-product formulas to numerator and denominator.</p><p>Numerator: sin x - sin z = 2cos((x+z)/2)sin((x-z)/2)</p><p>Denominator: cos z - cos x = -[cos x - cos z] = -[-2sin((x+z)/2)sin((x-z)/2)] = 2sin((x+z)/2)sin((x-z)/2)</p><p><strong>Step 2:</strong> Simplify the fraction.</p><p>cot y = [2cos((x+z)/2)sin((x-z)/2)] / [2sin((x+z)/2)sin((x-z)/2)]</p><p>cot y = cos((x+z)/2) / sin((x+z)/2) = cot((x+z)/2)</p><p><strong>Step 3:</strong> Conclude that y = (x+z)/2 (within appropriate domain).</p><p>∴ The possible answer is that y = (x+z)/2 or equivalently, y, x, z are in arithmetic progression (if x, z are angles of a triangle or specific configuration).</p><p>Answer: A</p>
Correct Answer: A