Find the value of the following inverse trigonometric expression:
`tan^(-1)(tan(-6))`

Explanation

To find the value of the expression \( \tan^{-1}(\tan(-6)) \), we will follow these steps:

### Step 1: Understand the properties of the inverse tangent function
The function \( \tan^{-1}(x) \) is defined for all real numbers \( x \), but it only returns values in the range \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \). Therefore, we need to find a corresponding angle for \( -6 \) that lies within this range.

### Step 2: Rewrite the angle
Since \( -6 \) is outside the range of \( \tan^{-1} \), we can add \( 2\pi \) to \( -6 \) to find an equivalent angle that lies within the range of the tangent function.

\[ -6 + 2\pi = -6 + 2 \times 3.14 \approx -6 + 6.28 = 0.28 \]

However, this is not necessary for our calculation, as we can also express \( -6 \) in terms of \( 2\pi \):

\[ -6 = 2\pi - (2\pi - 6) \]

### Step 3: Use the periodicity of the tangent function
Using the periodicity of the tangent function, we know that:

\[ \tan(2\pi - x) = -\tan(x) \]

Thus, we can express:

\[ \tan(-6) = \tan(2\pi - 6) \]

### Step 4: Apply the inverse tangent function
Now, we can rewrite the original expression:

\[ \tan^{-1}(\tan(-6)) = \tan^{-1}(\tan(2\pi - 6)) \]

Since \( 2\pi - 6 \) is in the range of \( \tan^{-1} \), we can directly evaluate:

\[ \tan^{-1}(\tan(2\pi - 6)) = 2\pi - 6 \]

### Step 5: Final answer
Thus, the value of \( \tan^{-1}(\tan(-6)) \) is:

\[ \boxed{2\pi - 6} \]