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To Find: The Principle value of $\operatorname{cosec}^{-1}(-\sqrt{2})$

Let the principle value be given by $x$

Now, let $x=\operatorname{cosec}^{-1}(-\sqrt{2})$

$\Rightarrow \operatorname{cosec} x=-\sqrt{2}$

$\Rightarrow \operatorname{cosec} x=-\operatorname{cosec}\left(\frac{\pi}{4}\right)\left(\because \operatorname{cosec}\left(\frac{\pi}{4}\right)=\sqrt{2}\right)$

$\Rightarrow \operatorname{cosec} x=\operatorname{cosec}\left(-\frac{\pi}{4}\right)(\because-\operatorname{cosec}(\theta)=\operatorname{cosec}(-\theta))$

$\Rightarrow x=-\frac{\pi}{4}$