Mark the tick against the correct answer in the following:
The principal value of $\operatorname{cosec}^{-1}(-\sqrt{2})$ is
A. $\frac{-\pi}{4}$
B. $\frac{3 \pi}{4}$
C. $\frac{5 \pi}{4}$
D. none of these
To Find: The Principle value of $\operatorname{cosec}^{-1}(-\sqrt{2})$
Let the principle value be given by $x$
Now, let $x=\operatorname{cosec}^{-1}(-\sqrt{2})$
$\Rightarrow \operatorname{cosec} x=-\sqrt{2}$
$\Rightarrow \operatorname{cosec} x=-\operatorname{cosec}\left(\frac{\pi}{4}\right)\left(\because \operatorname{cosec}\left(\frac{\pi}{4}\right)=\sqrt{2}\right)$
$\Rightarrow \operatorname{cosec} x=\operatorname{cosec}\left(-\frac{\pi}{4}\right)(\because-\operatorname{cosec}(\theta)=\operatorname{cosec}(-\theta))$
$\Rightarrow x=-\frac{\pi}{4}$
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