Math Doubts

Cos squared formula

Formula

$\cos^2{\theta} \,=\, 1-\sin^2{\theta}$

The square of cosine function equals to the subtraction of square of sin function from one is called the cosine squared formula. It is also called as the square of cos function identity.

Introduction

In trigonometric expressions and equations, the cosine functions are often involved in square form. The equations or expressions can be simplified only by transforming the square of cosine functions into its equivalent form. Therefore, it is very important to learn the square of cosine function formula for studying the trigonometry further.

Usage

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

  1. The square of cos function is expanded as the subtraction of sin squared function from one.
  2. The subtraction of sin squared function from one is simplified as the square of cos function.

Popular forms

The cosine squared function rule is also expressed popularly in two forms in mathematics.

  1. $\cos^2{x} \,=\, 1-\sin^2{x}$
  2. $\cos^2{A} \,=\, 1-\sin^2{A}$

Thus, you can express the square of cosine function formula in terms of any angle in trigonometry.

Proof

Let’s assume, theta is an angle of a right triangle, then the cosine and sine are written mathematically as $\cos{\theta}$ and $\sin{\theta}$ respectively. As per the Pythagorean identity of sin and cos functions, the mathematical relationship between cosine and sine can be written in the following mathematical form.

$\sin^2{\theta}+\cos^2{\theta} \,=\, 1$

$\therefore \,\,\,\,\,\, \cos^2{\theta} \,=\, 1-\sin^2{\theta}$

So, it is proved that the square of cosine function is equal to the subtraction of the square of sine function from one.