# Cotangent Function is Odd

## Theorem

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

Let $\cot x$ be the cotangent of $x$.

Then, whenever $\cot x$ is defined:

$\map \cot {-x} = -\cot x$

That is, the cotangent function is odd.

## Proof

 $\displaystyle \map \cot {-x}$ $=$ $\displaystyle \frac {\map \cos {-x} } {\map \sin {-x} }$ Cotangent is Cosine divided by Sine $\displaystyle$ $=$ $\displaystyle \frac {-\sin x} {\cos x}$ Cosine Function is Even and Sine Function is Odd $\displaystyle$ $=$ $\displaystyle -\cot x$ Cotangent is Cosine divided by Sine

$\blacksquare$