# Cosecant Function is Odd

## Theorem

Let $x \in \R$ be a real number.

Let $\csc x$ be the cosecant of $x$.

Then, whenever $\csc x$ is defined:

$\csc \left({-x}\right) = -\csc x$

That is, the cosecant function is odd.

## Proof

 $\displaystyle \csc \left({-x}\right)$ $=$ $\displaystyle \frac 1 {\sin \left({-x}\right)}$ Cosecant is Reciprocal of Sine $\displaystyle$ $=$ $\displaystyle \frac 1 {- \sin x}$ Sine Function is Odd $\displaystyle$ $=$ $\displaystyle - \csc x$ Cosecant is Reciprocal of Sine

$\blacksquare$