# Tangent Function is Odd

From ProofWiki

## Contents

## Theorem

For all $x \in \C$ where $\tan x$ is defined:

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

That is, the tangent function is odd.

## Proof

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \tan \left({-x}\right)\) | \(=\) | \(\displaystyle \) | \(\displaystyle \frac {\sin \left({-x}\right)} {\cos \left({-x}\right)}\) | \(\displaystyle \) | \(\displaystyle \) | Tangent is Sine divided by Cosine | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\displaystyle \frac {- \sin x} {\cos x}\) | \(\displaystyle \) | \(\displaystyle \) | Sine Function is Odd; Cosine Function is Even | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\displaystyle - \tan x\) | \(\displaystyle \) | \(\displaystyle \) | Tangent is Sine divided by Cosine |

$\blacksquare$

## Also see

- Sine Function is Odd
- Cosine Function is Even
- Cotangent Function is Odd
- Secant Function is Even
- Cosecant Function is Odd

## Sources

- Murray R. Spiegel:
*Theory and Problems of Complex Variables*(1964)... (previous)... (next): $2$: The Elementary Functions: $4$ - Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*(1968)... (previous)... (next): $\S 5$: Trigonometric Functions: $5.30$ - Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*(1968)... (previous)... (next): $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I