Answer
$-90^o$
Work Step by Step
The value of $\theta$ must be in the interval $[-90^o, 0^o) \cup (0^o, 90^o] $.
$\theta=\csc^{-1}{\left(-1\right)}$ means that $\csc{\theta} = -1$.
Since cosecant is the reciprocal of sine, then $\csc{\theta} = -1$ means $\sin{\theta}=-1$.
Note that $\sin{90^o} = 1$.
Since $\sin{(-x)}=-\sin{x}$, then
$\sin{(-90^o)}=-\sin{90^o} = -1$
Thus, $\csc^{-1}{\left(-1\right)}=-90^o$
