Pythagorean Identity

The Pythagorean Identity is a fundamental relation in trigonometry relating the square of the sine and cosine functions of an angle. It is derived from the Pythagorean theorem when applied to the unit circle[1]. The identity can also be visualized as the two lengths that divide the radius into two smaller right triangles[2], as shown below.

Pythagorean Identity
Note: The notation is shorthand for . This is important because in other places in math, the notation can mean something else when referring to a generic function .


Derive Pythagorean Identity

To derive the Pythagorean identity the definition of Pythagorean's theorem is combined with the notion of a right triangle placed on the unit circle.

Derive Pythagorean Identity (Unit Circle)

To geometrically derive the Pythagorean identity, divide the right triangle into two similar triangles by drawing a line from the right angle corner of the right-triangle perpendicular to its hypotenuse. Then solve for each triangle's length on the hypotenuse.


  1. Derive Pythagorean Identity Example
  2. Derive Pythagorean Identity (Unit Circle) Example