Answer
True
Work Step by Step
Figure 40 shows that the graph of $y=csc{x}$ is symmetric with respect to the origin.
It can be seen from the figure that:
(i) $\csc{(-\frac{3\pi}{2})}= 1$ while $\csc{(\frac{3\pi}{2})}=-1$
(ii) $\csc{(-\frac{\pi}{2})}=-1$ while $\csc{(\frac{\pi}{2})}=1$
In general, $\csc{(-x)}=-\csc{x}$.
$\csc{x} = \sec{(-x)}$.
Another way showing that $\csc{(-x)}=-\csc{x}$ is by using the concept of odd functions.
Recall that when the graph of a function is symmetric with respect to the origin, then, the function is an odd function. If a function is odd, then $f(-x)=-f(x)$.
Thus, since $y=\csc{x}$ is an odd function, then $\csc{(-x)}=-\csc{x}$ for all numbers within its domain.
