${\cos ^{ - 1}}\left( { - \frac{1}{{\sqrt 2 }}} \right)$

Let ${\cos ^{ - 1}}\left( { - \frac{1}{{\sqrt 2 }}} \right) = x \Rightarrow - \frac{1}{{\sqrt 2 }} = \cos x$

As we know that the range of principal value branch of ${\cos ^{ - 1}}is\;[0,\;\pi ]$

Then, $- \frac{1}{{\sqrt 2 }} = - \cos \left( {\frac{\pi }{4}} \right) = \cos \left( {\pi - \frac{\pi }{4}} \right) = \cos \frac{{3\pi }}{4},$

Where $\frac{{3\pi }}{4} \in [0,\;\pi ]$

Hence, the principal value of ${\cos ^{ - 1}}\left( { - \frac{1}{{\sqrt 2 }}} \right)$ is $\frac{{3\pi }}{4}.$