What is 2\cot\left(\frac{1}{2}\cos^{-1}\frac{\sqrt{5}}{3}\right) equal to?
- A. -1
- B. 1
- C. 3+\sqrt{5} ✓
- D. 3-\sqrt{5}
Correct Answer: C. 3+\sqrt{5}
Explanation
Let \theta = \cos^{-1}\frac{\sqrt{5}}{3}, so \cos\theta = \frac{\sqrt{5}}{3}. From the half-angle formula, \cot\left(\frac{\theta}{2}\right) = \frac{1 + \cos\theta}{\sin\theta}. Since \sin\theta = \sqrt{1 - \cos^2\theta} = \frac{2}{3}, this evaluates to \frac{1 + \sqrt{5}/3}{2/3} = \frac{3+\sqrt{5}}{2}. Multiplying by 2 gives 3+\sqrt{5}.