What is pq equal to ?
Consider the following for the next two (02) items that follow : Let p=\frac{1}{3}-\frac{\tan 3x}{\tan x} and q=1-3\tan^{2}x, 0 \lt x \lt \pi, x \neq \frac{\pi}{2}.
- A. 1
- B. 2
- C. \frac{8}{3}
- D. -\frac{8}{3} ✓
Correct Answer: D. -\frac{8}{3}
Explanation
Expanding \tan 3x gives p = \frac{1}{3} - \frac{3\tan x - \tan^3 x}{\tan x (1 - 3\tan^2 x)} = \frac{1}{3} - \frac{3-\tan^2 x}{1-3\tan^2 x}. Simplifying this yields p = \frac{1-3\tan^2 x - 9 + 3\tan^2 x}{3(1-3\tan^2 x)} = \frac{-8}{3(1-3\tan^2 x)}. Notice that q = 1-3\tan^2 x. Thus pq = \frac{-8}{3(1-3\tan^2 x)} \times (1-3\tan^2 x) = -\frac{8}{3}.