Find the principal value of $\sec^{-1} \left(- \sqrt{2}\right)$
Let $\;$ $\sec^{-1} \left(- \sqrt{2}\right) = y$, $\;$ where $\;$ $0 < y \leq \pi; \;\; y \neq \dfrac{\pi}{2}$
$\implies$ $\sec \left(y\right) = - \sqrt{2} = \sec \left(\dfrac{3 \pi}{4}\right)$
$\implies$ $y = \dfrac{3 \pi}{4}$
$\therefore \;$ The principal value of $\sec^{-1} \left(- \sqrt{2}\right)$ is $\dfrac{3 \pi}{4}$