Cot squared formula

Formula

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

The square of cot function equals to the subtraction of one from the square of co-secant function is called the cot squared formula. It is also called as the square of cot function identity.

Introduction

The cotangent functions are sometimes appeared in square form in trigonometric expressions and equations. Actually, the expressions or equations can be simplified possibly by transforming the cot squared functions into its equivalent form. Hence, it is must to learn the square of cot function rule for studying the trigonometry further.

Usage

The cotangent squared trigonometric identity is used as a formula in two cases.

The square of cot function is expanded as the subtraction of one from the co-secant squared function.
The subtraction of one from the cosecant squared function is simplified as the square of co-tangent function.

Popular forms

In trigonometry, the cotangent squared function rule is also popularly expressed in two forms.

$\cot^2{x} \,=\, \csc^2{x}-1$
$\cot^2{A} \,=\, \csc^2{A}-1$

Mathematically, you can write the square of cotangent function law in terms of any angle in the same way.

Proof

Assume, theta is an angle of a right triangle, then the cotangent and cosecant are written as $\cot{\theta}$ and $\csc{\theta}$ respectively in trigonometric mathematics. The relationship between cot and cosecant functions can be written in the following mathematical form as per the Pythagorean identity of cot and cosecant functions.

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

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

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

Therefore, it has derived that the square of cot function is equal to the subtraction one from the square of cosecant function.