# Tangent Function is Odd

## Theorem

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

Let $\tan x$ be the tangent of $x$.

Then, whenever $\tan x$ is defined:

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

That is, the tangent function is odd.

## Proof

 $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle \tan \left({-x}\right)$$ $$=$$ $$\displaystyle$$ $$\displaystyle \frac {\sin \left({-x}\right)} {\cos \left({-x}\right)}$$ $$\displaystyle$$ $$\displaystyle$$ Definition of tangent $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \frac {- \sin x} {\cos x}$$ $$\displaystyle$$ $$\displaystyle$$ Sine Function is Odd; Cosine Function is Even $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle - \tan x$$ $$\displaystyle$$ $$\displaystyle$$ Definition of tangent

$\blacksquare$