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The definition of the cosecant function is
[tex]\csc \theta=\frac{1}{\sin \theta}[/tex]Therefore,
[tex]\Rightarrow\csc (-\frac{\pi}{2})=\frac{1}{\sin (-\frac{\pi}{2})}[/tex]To find sin(-pi/2), use the diagram below.
Consider that the circumference has a radius equal to 1. Then, the coordinates of the orange point are (0,-1). Furthermore, the points on the circumference are given as (cos(theta), sin(theta)); therefore,
[tex]\begin{gathered} \Rightarrow(0,-1)=(\cos (-\frac{\pi}{2}),\sin (-\frac{\pi}{2})) \\ \Rightarrow\sin (-\frac{\pi}{2})=-1 \\ \Rightarrow\csc (-\frac{\pi}{2})=\frac{1}{-1}=-1 \\ \Rightarrow\csc (-\frac{\pi}{2})=-1 \end{gathered}[/tex]Thus, the answer is csc(-pi/2)=-1
