Question

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

Answer 1

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