Answer
The exact value of the provided expression is $\sqrt{2}$.
Work Step by Step
We know that the reference angle for $\theta =\frac{7\pi }{4}$ is:
$\begin{align}
& \theta '=2\pi -\frac{7\pi }{4} \\
& =\frac{8\pi -7\pi }{4} \\
& =\frac{\pi }{4}
\end{align}$
Now, the function value for the reference angle is:
$\sec \frac{\pi }{4}=\sqrt{2}$
Thus, we see that the angle $\frac{7\pi }{4}$ lies in quadrant IV, so the sec function is positive:
$\begin{align}
& \sec \frac{7\pi }{4}=\sec \frac{\pi }{4} \\
& =\sqrt{2}
\end{align}$
