# Tangent Function is Odd

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## Contents

## Theorem

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

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

That is, the tangent function is odd.

## Proof

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

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

\(\displaystyle \) | \(=\) | \(\displaystyle -\tan x\) | 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

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.30$ - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $4$