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Question

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

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Answer

Let $\cos ^{-1}\left(\frac{-1}{\sqrt{2}}\right)=y$
$\Rightarrow \cos y=-\frac{1}{\sqrt{2}}$
$\Rightarrow \cos y=-\cos \frac{\pi}{4}$
$\Rightarrow \cos y=\cos \left(\pi-\frac{\pi}{4}\right)=\cos \frac{3 \pi}{4}$
Since, the principal value branch of $\cos ^{-1}$ is $[0, \pi]$.
Therefore, Principal value of $\cos ^{-1}\left(\frac{-1}{\sqrt{2}}\right)$ is $\frac{3 \pi}{4}$.

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