Chapter 6 - Analytic Trigonometry - Section 6.2 The Inverse Trigonometric Functions (Continued) - 6.2 Assess Your Understanding - Page 480: 39

$\dfrac{-\pi}{2}$

We know that the trigonometric function $\csc^{-1} x$ has range: $[-\dfrac{\pi}{2}, 0) \cup (0, \dfrac{\pi}{2}]$. Here, $\csc^{-1} (-1)=\dfrac{-\pi}{2}$, because $\csc(-\dfrac{\pi}{2})=\dfrac{-1}{1}=-1$ and $\dfrac{-\pi}{2}$ lies in the range of $\csc^{-1}{x}$ .