Answer
The value of $y$ here is $$y=\frac{3\pi}{4}$$
Work Step by Step
$$y=\sec^{-1}(-\sqrt 2)$$
First, we see that the domain of inverse secant function is $(-\infty,\infty)$. Therefore, in fact when we deal with inverse secant function, we do not need to do this checking step.
The range of inverse secant function is $[0,\frac{\pi}{2})\hspace{0.2cm}U\hspace{0.2cm}(\frac{\pi}{2},\pi]$. In other words, $y\in[0,\frac{\pi}{2})\hspace{0.2cm}U\hspace{0.2cm}(\frac{\pi}{2},\pi]$.
We can rewrite $y=\sec^{-1}(-\sqrt 2)$ into $\sec y=-\sqrt2$
We know that $$\sec\frac{3\pi}{4}=\frac{1}{\cos\frac{3\pi}{4}}=\frac{1}{-\frac{\sqrt 2}{2}}=-\frac{2}{\sqrt2}=-\sqrt2$$
And $\frac{3\pi}{4}$ belongs to the range $[0,\frac{\pi}{2})\hspace{0.2cm}U\hspace{0.2cm}(\frac{\pi}{2},\pi]$.
Therefore, the exact value of $y$ here is $$y=\frac{3\pi}{4}$$
