Question

Find the principal value of cosec-1$(-\sqrt2)$

Answer 1

Let cosec-1$(-\sqrt2)=y.$
Then $cosec\, y =-\sqrt2=-cosec\bigg(\frac{\pi}{4}\bigg)=cosec\bigg(-\frac{\pi}{4}\bigg).$
We know that the range of the principal value branch of cosec-1 is
$\bigg[-\frac{\pi}{2},\frac{\pi}{2}\bigg]-\\{0\\} and \, cosec\bigg(-\frac{\pi}{4}\bigg)=-\sqrt2.$
Therefore, the principal value of cosec-1$(-\sqrt2)$ is $-\frac{\pi}{4}.$