Find the principal value of cosec^ -1 (-2). - Vidyadip

Question

Find the principal value of $cosec^{-1}(-\sqrt{2})$.

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Answer

Let $cosec^{-1}(-\sqrt{2})=y$. Then, $cosec\ y = -\sqrt{2}=-cosec \left(\frac{\pi}{4}\right)=cosec \left(-\frac{\pi}{4}\right)$.
We know that the range of the principal value branch of
$cosec^{-1}$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]-\{0\}$ and $cosec \left(-\frac{\pi}{4}\right)=-\sqrt{2}$.
Therefore, the principal value of $cosec^{-1}(-\sqrt{2})$ is $-\frac{\pi}{4}$.

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