Answer
$-\sqrt {2}$
Work Step by Step
$\csc(-855^{\circ})=\frac{1}{\sin (-855^{\circ})}$
Let us find $\sin (-855^{\circ})$ first.
As the period of sine function is $360^{\circ}$,
$\sin(-855^{\circ})=\sin(-855^{\circ}+(2\times360^{\circ}))$
$=\sin(-135^{\circ})=-\sin 135^{\circ}$
(As $\sin (-x)=-\sin x$ )
$=-\sin(180^{\circ}-45^{\circ})=-\sin 45^{\circ}$
( As $\sin(180^{\circ}-x)=\sin x$ )
$=-\frac{1}{\sqrt 2}$
$\csc(-855^{\circ})=\frac{1}{-\frac{1}{\sqrt 2}}=-\sqrt 2$
