<p>Find the value of \(\frac{\cos A + \cos B}{\sin A - \sin B} + \frac{\sin A + \sin B}{\cos A - \cos B}\) (where \(n\) is even).</p>
Step-by-Step Solution
Key Concept: Use sum-to-product formulas to convert sums and differences of sines and cosines into products, then simplify the resulting expression by finding a common denominator or recognizing trigonometric identities.
<p><strong>Step 1: Apply sum-to-product formulas</strong></p><p>Recall the formulas:</p><p>• cos A + cos B = 2cos((A+B)/2)cos((A-B)/2)</p><p>• sin A - sin B = 2cos((A+B)/2)sin((A-B)/2)</p><p>• sin A + sin B = 2sin((A+B)/2)cos((A-B)/2)</p><p>• cos A - cos B = -2sin((A+B)/2)sin((A-B)/2)</p><p><strong>Step 2: Substitute into first fraction</strong></p><p>$$\frac{\cos A + \cos B}{\sin A - \sin B} = \frac{2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)}{2\cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)}$$</p><p>Simplifying (assuming cos((A+B)/2) ≠ 0):</p><p>$$= \frac{\cos\left(\frac{A-B}{2}\right)}{\sin\left(\frac{A-B}{2}\right)} = \cot\left(\frac{A-B}{2}\right)$$</p><p><strong>Step 3: Substitute into second fraction</strong></p><p>$$\frac{\sin A + \sin B}{\cos A - \cos B} = \frac{2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)}{-2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)}$$</p><p>Simplifying (assuming sin((A+B)/2) ≠ 0):</p><p>$$= \frac{\cos\left(\frac{A-B}{2}\right)}{-\sin\left(\frac{A-B}{2}\right)} = -\cot\left(\frac{A-B}{2}\right)$$</p><p><strong>Step 4: Add the two fractions</strong></p><p>$$\cot\left(\frac{A-B}{2}\right) + \left(-\cot\left(\frac{A-B}{2}\right)\right) = \cot\left(\frac{A-B}{2}\right) - \cot\left(\frac{A-B}{2}\right) = 0$$</p><p><strong>∴ Answer: 0</strong></p>
Correct Answer: 0