Pythagorean identities

A trigonometric equation that expresses the Pythagorean Theorem is called the Pythagorean identity.

Introduction

In trigonometry, a trigonometric function has a mathematical relation with another trigonometric function in some special forms. Surprisingly, two trigonometric functions in square form express the Pythagorean Theorem. Hence, the mathematical relationship between them is called the Pythagorean identity.

The six trigonometric functions possibly express the Pythagoras Theorem in the following three forms. If you are a beginner, you must learn all of them for studying trigonometry. So, let’s learn each Pythagorean identity with proof.

Sine and Cosine functions

The sum of the squares of the sine and cosine functions is equal to one.

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

Secant and Tangent functions

The subtraction of the square of tan function from square of secant function is equal to one.

$\sec^2{\theta} \,-\, \tan^2{\theta} \,=\, 1$

Cosecant and Cotangent functions

The subtraction of the square of cotangent function from square of cosecant function is equal to one.

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

In some countries, this pythagorean identity is also written in the following form.

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

Usage

The Pythagorean identities are mainly used as formulas to express one trigonometric function in terms of another trigonometric function.