Pythagorean identities

Definition

A trigonometric identity that represents Pythagorean Theorem in terms of trigonometric functions is called Pythagorean trigonometric identity or simply Pythagorean identity.

right angled triangle

Pythagorean Theorem expresses the relation between lengths of three sides of a right angled triangle in square form.

The same theorem can be transformed in ratios of two sides of the right angled triangle and then the associated ratios can be replaced by the trigonometric functions to get the Pythagorean theorem in terms of trigonometric functions.

Pythagorean identities List

The six trigonometric functions can represent Pythagorean Theorem in three different forms. For example, If theta ($\theta$) is an angle of the right angled triangle, The Pythagorean theorem can be expressed in terms of trigonometric functions in the following three ways.

1

Sine and Cosine

$\sin^2 \theta + \cos^2 \theta = 1$

The identity is read as sin squared theta plus cos squared theta is equal to one.

The relation between sine and cosine functions exactly expresses the Pythagorean Theorem. Hence, the Pythagorean relation between sine and cosine is called the fundamental Pythagorean trigonometric identity.

Learn the proof of Pythagorean relation between sine and cosine functions.

2

Secant and Tangent

$\sec^2 \theta -\tan^2 \theta = 1$

It is read as secant squared theta minus tan squared theta is equal to one.

The relation between secant and tangent in square form also represents the Pythagoras theorem and learn the proof of Pythagorean trigonometric identity of secant and tangent.

3

Cosecant and Cotangent

$\csc^2 \theta -\cot^2 \theta = 1$

It is read as cosecant squared theta minus cot squared theta is equal to one.

The trigonometric functions cosecant and cotangent also express the Pythagorean theorem in square form and learn the proof of Pythagorean identity of cosecant and cotangent.

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