What is \( \tan\left[2\tan^{-1}\frac{1}{2} - \frac{\pi}{4}\right] \) equal to ?
- A. -7
- B. 0
- C. \frac{1}{5}
- D. \frac{1}{7} ✓
Correct Answer: D. \frac{1}{7}
Explanation
Let \alpha = \tan^{-1}(1/2). Then 2\alpha = \tan^{-1}(2(1/2)/(1-1/4)) = \tan^{-1}(4/3). Thus \tan(2\alpha) = 4/3. Applying the formula \tan(x - y) = (\tan x - \tan y)/(1 + \tan x \tan y), we get (4/3 - 1)/(1 + 4/3) = (1/3)/(7/3) = 1/7.