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

Explanation

To find the value of the expression \( \cos^{-1}(\cos 10) \), we can follow these steps:

### Step 1: Identify the angle
We need to evaluate \( \cos^{-1}(\cos 10) \). The angle \( 10 \) is in radians.

### Step 2: Determine the equivalent angle in the principal range
The principal range of the inverse cosine function \( \cos^{-1}(x) \) is \( [0, \pi] \). Since \( 10 \) radians is greater than \( \pi \) (approximately \( 3.14 \)), we need to find an equivalent angle within the range of \( [0, \pi] \).

### Step 3: Find the equivalent angle
To find the equivalent angle, we can subtract \( 2\pi \) (or multiples of \( 2\pi \)) from \( 10 \) until we get an angle within the principal range.

Calculating:
\[ 10 - 2\pi \approx 10 - 6.28 \approx 3.72 \]
This angle \( 3.72 \) is still greater than \( \pi \).

Now, let's subtract \( 2\pi \) again:
\[ 3.72 - 2\pi \approx 3.72 - 6.28 \approx -2.56 \]
This angle is less than \( 0 \).

Now, let's try subtracting \( \pi \) from \( 10 \):
\[ 10 - \pi \approx 10 - 3.14 \approx 6.86 \]
This angle is still greater than \( \pi \).

Now, we can check:
\[ 10 - 3\pi \approx 10 - 9.42 \approx 0.58 \]
This angle \( 0.58 \) is within the range \( [0, \pi] \).

### Step 4: Evaluate the expression
Now we can write:
\[ \cos^{-1}(\cos 10) = \cos^{-1}(\cos(3\pi + \delta)) \]
Where \( \delta = 10 - 3\pi \).

Using the property of cosine:
\[ \cos(3\pi + \delta) = -\cos(\delta) \]
Thus,
\[ \cos^{-1}(\cos(3\pi + \delta)) = \pi - \delta \]
So,
\[ \cos^{-1}(\cos 10) = \pi - (10 - 3\pi) = 4\pi - 10 \]

### Final Result
The value of \( \cos^{-1}(\cos 10) \) is \( 4\pi - 10 \).