Evaluate: `tan{1/2cos^(-1)(sqrt5/3)}`

Explanation

To evaluate \( \tan\left(\frac{1}{2} \cos^{-1}\left(\frac{\sqrt{5}}{3}\right)\right) \), we can follow these steps:

### Step 1: Let \( x = \cos^{-1}\left(\frac{\sqrt{5}}{3}\right) \)
This implies that \( \cos x = \frac{\sqrt{5}}{3} \).

### Step 2: Use the half-angle formula for tangent
The half-angle formula for tangent is given by:
\[
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