Question

How do you know if `csc x` is an even or odd function?

Answer 1

Odd.

Explanation:

Know that `sin(-x)=-sin(x)`.

Even function: when `f(-x)=f(x)`
Odd function: when `f(-x)=-f(x)`

In this case, `f(x)=cscx`

`f(x)=1/sinx`

`f(-x)=1/(sin(-x))`

`f(-x)=1/(-sinx)`

`f(-x)=-cscx=-f(x)`

Thus, the function is odd.
graph{csc(x) [-10, 10, -5, 5]}
A property of odd functions is that they have origin symmetry, which means the graph can is symmetrical when reflected over the point `(0,0)`.