What is the value of \tan\left[\frac{1}{2}\sec^{-1}\left(\frac{2}{\sqrt{3}}\right)\right]?
- A. 2-\sqrt{3} ✓
- B. 2+\sqrt{3}
- C. \sqrt{3}-1
- D. \sqrt{3}+1
Correct Answer: A. 2-\sqrt{3}
Explanation
Evaluate the inner function: \sec^{-1}\left(\frac{2}{\sqrt{3}}\right) = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6}. The expression becomes \tan\left(\frac{1}{2} \times \frac{\pi}{6}\right) = \tan\left(\frac{\pi}{12}\right), which is \tan(15^\circ). Using the half-angle or subtraction formula, \tan(15^\circ) = 2 - \sqrt{3}.