Find the principal value of cosec^{-1}(-sqrt{ | vedclass

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(D) Let $cosec^{-1}(-\sqrt{2}) = y$. Then,$cosec\; y = -\sqrt{2}$. We know that $cosec\left(\frac{\pi}{4}\right) = \sqrt{2}$,so $cosec\left(-\frac{\pi

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$-\frac{\pi}{4}$

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

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

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