Evaluate the following expression:
`tan("cos"^(-1)1/3)`

Explanation

To evaluate the expression \( \tan(\cos^{-1}(\frac{1}{3})) \), we will follow these steps:

### Step 1: Set up the equation
Let \( \theta = \cos^{-1}(\frac{1}{3}) \). This means that \( \cos(\theta) = \frac{1}{3} \).

**Hint:** Remember that \( \cos^{-1}(x) \) gives you an angle whose cosine is \( x \).

### Step 2: Use the Pythagorean identity
From the definition of cosine, we can use the Pythagorean identity to find the sine of \( \theta \):
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
Substituting \( \cos(\theta) \):
\[ \sin^2(\theta) + \left(\frac{1}{3}\right)^2 = 1 \]
\[ \sin^2(\theta) + \frac{1}{9} = 1 \]
\[ \sin^2(\theta) = 1 - \frac{1}{9} = \frac{8}{9} \]
Thus, \( \sin(\theta) = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3} \).

**Hint:** Use the Pythagorean identity to relate sine and cosine.

### Step 3: Find \( \tan(\theta) \)
Now that we have both sine and cosine, we can find \( \tan(\theta) \):
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{2\sqrt{2}}{3}}{\frac{1}{3}} = 2\sqrt{2} \]

**Hint:** Recall that \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

### Step 4: Conclude the evaluation
Thus, the value of \( \tan(\cos^{-1}(\frac{1}{3})) \) is:
\[ \tan(\cos^{-1}(\frac{1}{3})) = 2\sqrt{2} \]

**Final Answer:** \( 2\sqrt{2} \)