# Cot Double angle formula

### Expansion form

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

### Simplified form

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

### Introduction

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

1. Cot of double angle is expanded as the quotient of subtraction of one from square of cot function by twice the cot function.
2. The quotient of subtraction of one from square of cot function by twice the cot function is simplified as cot of double angle.

#### How to use

The co-tangent of double angle identity is used to either expand or simplify the double angle functions like $\cot{2A}$, $\cot{2x}$, $\cot{2\alpha}$ and etc. For example,

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

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

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

#### Proof

Learn how to derive the rule of cot double angle identity by geometric approach in trigonometry.

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Jun 26, 2023

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