# Cotangent Function is Odd

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## Contents

## 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 |

$\blacksquare$

## Also see

- Sine Function is Odd
- Cosine Function is Even
- Tangent 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: $5.33$ - 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