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

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

$\blacksquare$


Examples

Cotangent of $-2 x$

$\map \cot {-2 x} = -\cot 2 x$


Also see


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
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