What is the value of p+q?
Consider the following for the next two (02) items that follow : Let p=\cos\left(\frac{\pi}{5}\right)\cos\left(\frac{2\pi}{5}\right) and q=\cos\left(\frac{4\pi}{5}\right)\cos\left(\frac{8\pi}{5}\right).
- A. -\frac{1}{2}
- B. -\frac{1}{4}
- C. 0 ✓
- D. \frac{1}{2}
Correct Answer: C. 0
Explanation
Note that \cos(\pi/5) = \cos 36^\circ = \frac{\sqrt{5}+1}{4} and \cos(2\pi/5) = \cos 72^\circ = \frac{\sqrt{5}-1}{4}. Thus p = \frac{5-1}{16} = \frac{1}{4}. Using allied angles, q = \cos(\pi - \pi/5)\cos(2\pi - 2\pi/5) = (-\cos(\pi/5))(\cos(2\pi/5)) = -p = -\frac{1}{4}. Their sum p+q = \frac{1}{4} - \frac{1}{4} = 0.