What is \frac{p^2-q^2}{p^2+q^2} equal to?
For the following two (02) items: Let \sin A + \sin B = p and \cos A + \cos B = q.
- A. \cos(A+B)
- B. \cos(A-B)
- C. \cos(\frac{\pi}{2}-A-B)
- D. \cos(\pi-A-B) ✓
Correct Answer: D. \cos(\pi-A-B)
Explanation
Using the formulas for p and q, p^2+q^2 = 4\cos^2(\frac{A-B}{2}) and p^2-q^2 = 4\cos^2(\frac{A-B}{2}) [\sin^2(\frac{A+B}{2}) - \cos^2(\frac{A+B}{2})] = -4\cos^2(\frac{A-B}{2})\cos(A+B). Thus, the ratio is -\cos(A+B) = \cos(\pi-A-B).