Answer
Cosine and secant are positive. All others are negative.
Work Step by Step
A $-80^{\circ}$ is in quadrant IV. In the fourth quadrant, $x$ and $r$ are positive but $y$ is negative.
$\implies \sin\theta=\frac{y}{r}=\text{negative}$
As $\csc\theta=\frac{1}{\sin\theta}$, $\csc\theta$ is also negative.
$\cos\theta=\frac{x}{r}=\text{positive}$
Secant which is the inverse of cosine is also positive.
$\tan\theta=\frac{y}{x}=\text{negative}$
$\cot\theta=\frac{x}{y}=\text{negative}$
