Question

The value of \(\tan\left(\frac{1}{2}\cos^{-1}\left(\frac{\sqrt{5}}{3}\right)\right)\) is

• A. \(\frac{3-\sqrt{5}}{2}\)
• B. \(\frac{3+\sqrt{5}}{2}\)
• C. \(\frac{1}{2}(3-\sqrt{5})\)
• D. \(\frac{1}{2}(3+\sqrt{5})\)

Hint:

Use half-angle formula: \(\tan(\theta/2) = \sqrt{(1-\cos\theta)/(1+\cos\theta)}\).

Answer 1

The Correct Option is C

Step 1: Formula / Definition}
\[ \tan \frac{\theta}{2} = \sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}} \]
Step 2: Calculation / Simplification}
Let \(\theta = \cos^{-1}\left(\frac{\sqrt{5}}{3}\right) \Rightarrow \cos\theta = \frac{\sqrt{5}}{3}\)
\(\tan \frac{\theta}{2} = \sqrt{\frac{1 - \sqrt{5}/3}{1 + \sqrt{5}/3}} = \sqrt{\frac{3-\sqrt{5}}{3+\sqrt{5}}} = \sqrt{\frac{(3-\sqrt{5})^2}{9-5}} = \frac{3-\sqrt{5}}{2}\)
Step 3: Final Answer
\[ \frac{1}{2}(3-\sqrt{5}) \]