Cotangent Function is Odd
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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
- 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