# Cot triple angle formula

### Expansion form

$\cot{3\theta} \,=\, \dfrac{3\cot{\theta}-\cot^3{\theta}}{1-3\cot^2{\theta}}$

### Simplified form

$\dfrac{3\cot{\theta}-\cot^3{\theta}}{1-3\cot^2{\theta}} \,=\, \cot{3\theta}$

### Introduction

It is called cot triple angle identity and used in two different cases as a formula.

1. Cot of triple angle is expanded as the quotient of subtraction of cot cubed of angle from three times cot of angle by subtraction of three times cot squared of angle from one.
2. The quotient of subtraction of cot cubed of angle from three times cot of angle by subtraction of three times cot squared of angle from one is simplified as cot of triple angle.

#### How to use

The co-tangent of triple angle identity is used to either expand or simplify the triple angle cot functions like $\cot{3A}$, $\cot{3x}$, $\cot{3\alpha}$ and etc. For example,

$(1) \,\,\,\,\,\,$ $\cot{3x} \,=\, \dfrac{3\cot{x}-\cot^3{x}}{1-3\cot^2{x}}$

$(2) \,\,\,\,\,\,$ $\cot{3A} \,=\, \dfrac{3\cot{A}-\cot^3{A}}{1-3\cot^2{A}}$

$(3) \,\,\,\,\,\,$ $\cot{3\alpha} \,=\, \dfrac{3\cot{\alpha}-\cot^3{\alpha}}{1-3\cot^2{\alpha}}$

#### Proof

Learn how to derive the rule of cot triple angle identity in trigonometry by geometry.

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Jun 26, 2023
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