Math Doubts

Cosecant squared formula

Formula

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

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

Introduction

Sometimes, the cosecant functions are appeared in square form in trigonometric expressions and equations. The trigonometric expressions and equations can be simplified only by transforming the cosecant squared functions into its equivalent form. So, it is essential to learn the square of cosecant function identity in order to study the advanced trigonometry.

Usage

The cosecant squared trigonometric rule is mainly used as a formula in two cases.

  1. The square of cosecant function is expanded as the sum of one and the cotangent squared function.
  2. The summation of one and the co-tangent squared function is simplified as the square of cosecant function.

Popular forms

The cosecant squared function identity is also expressed in two forms popularly in trigonometry.

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

So, you can write the square of cosecant function rule in terms of any angle in this same way in trigonometric mathematics.

Proof

Let’s take, the theta is an symbol and it represents an angle of a right triangle. In mathematics, the cosecant and cotangent functions are written as $\csc{\theta}$ and $\cot{\theta}$ respectively. Now, the relationship between cosecant and cot functions can be written in the following mathematical form according to the Pythagorean identity of cosecant and cot functions.

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

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

Therefore, it has successfully derived that the square of cosecant function is equal to the addition of one and square of cot function.

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