Find the value of the following inverse trigonometric expression:
`cot^(-1)(cot(-10))`

Explanation

To solve the expression \( \cot^{-1}(\cot(-10)) \), we will follow these steps:

### Step 1: Understand the Inverse Cotangent Function
The function \( \cot^{-1}(x) \) gives us an angle whose cotangent is \( x \). However, this function is defined only for angles in the range \( (0, \pi) \).

### Step 2: Identify the Argument
We have \( \cot^{-1}(\cot(-10)) \). Here, the argument is \( -10 \).

### Step 3: Adjust the Angle to the Correct Range
Since \( -10 \) is not in the range \( (0, \pi) \), we need to adjust it. We can do this by adding a multiple of \( \pi \) to bring it into the desired range.

To find a suitable multiple of \( \pi \):
- We can add \( 4\pi \) to \( -10 \) because \( 4\pi \) is a full rotation (or multiple of \( 2\pi \)) plus some extra.

Calculating:
\[ 4\pi - 10 \]

### Step 4: Check the Resulting Angle
Now we need to check if \( 4\pi - 10 \) lies within the range \( (0, \pi) \).

Calculating \( 4\pi \):
\[ 4\pi \approx 4 \times 3.14 = 12.56 \]
Now, subtracting \( 10 \):
\[ 12.56 - 10 = 2.56 \]

Since \( 2.56 \) lies within \( (0, \pi) \), we can proceed.

### Step 5: Apply the Inverse Cotangent Identity
Now we can apply the identity:
\[ \cot^{-1}(\cot(x)) = x \quad \text{for } x \in (0, \pi) \]
Thus:
\[ \cot^{-1}(\cot(-10)) = 4\pi - 10 \]

### Final Answer
The value of \( \cot^{-1}(\cot(-10)) \) is:
\[ \boxed{4\pi - 10} \]