Evaluate the following inverse trigonometric expression:
`sec^(-1)("sec"(7pi)/4)`

Explanation

To evaluate the expression \( \sec^{-1}(\sec(7\pi/4)) \), we can follow these steps:

### Step 1: Understand the Inverse Secant Function
The function \( \sec^{-1}(x) \) is defined such that if \( y = \sec^{-1}(x) \), then \( x = \sec(y) \) and \( y \) is in the range \( [0, \pi] \) excluding \( \frac{\pi}{2} \).

### Step 2: Identify the Angle
We have \( \sec(7\pi/4) \). First, we need to find the angle \( 7\pi/4 \) in the standard range of the secant function. The angle \( 7\pi/4 \) is equivalent to \( 2\pi - \frac{\pi}{4} \) or \( -\frac{\pi}{4} \) when considering the unit circle.

### Step 3: Modify the Angle
Since \( 7\pi/4 \) does not lie in the range \( [0, \pi] \), we can modify it:
\[ 7\pi/4 = 2\pi - \frac{\pi}{4} = \frac{8\pi}{4} - \frac{\pi}{4} = \frac{7\pi}{4} \]
Thus, we can express it as:
\[ \sec(7\pi/4) = \sec(2\pi - \frac{\pi}{4}) = \sec(\frac{\pi}{4}) \]

### Step 4: Apply the Inverse Secant Property
Now we can write:
\[ \sec^{-1}(\sec(7\pi/4)) = \sec^{-1}(\sec(\frac{\pi}{4})) \]
Since \( \frac{\pi}{4} \) lies within the acceptable range \( [0, \pi] \), we can apply the property:
\[ \sec^{-1}(\sec(\theta)) = \theta \quad \text{for } \theta \in [0, \pi] \text{ and } \theta \neq \frac{\pi}{2} \]
Thus, we have:
\[ \sec^{-1}(\sec(7\pi/4)) = \frac{\pi}{4} \]

### Final Answer
Therefore, the value of \( \sec^{-1}(\sec(7\pi/4)) \) is:
\[ \frac{\pi}{4} \]