Answer
$\csc t=-\sqrt{1 +\cot^{2}t}$
Work Step by Step
see: Signs of the Trigonometric Functions (p.412)
$\left[\begin{array}{lll}
Quadrant & Positive & Negative\\
I & all & none\\
II & sin, csc & cos, sec, tan, cot\\
III & tan, cot & sin, csc, cos, sec\\
IV & cos, sec & sin, csc, tan, cot
\end{array}\right]$
also, on p.415, Reciprocal Identities:
$\displaystyle \csc t=\frac{1}{\sin t} \qquad \displaystyle \sec t=\frac{1}{\cos t}\qquad \displaystyle \cot t=\frac{1}{\tan t}$
and: Pythagorean Identities:
$\sin^{2}t+\cos^{2}t=1\qquad\tan^{2}t+1=\sec^{2}t\qquad 1 +\cot^{2}t=\csc^{2}t$
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$1 +\cot^{2}t=\csc^{2}t\qquad\Rightarrow \csc t=\pm\sqrt{1 +\cot^{2}t}$
By the sign table, Q.III: $\qquad \csc t=-\sqrt{1 +\cot^{2}t}$