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.
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.
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 righttriangle perpendicular to its hypotenuse. Then solve for each triangle's length on the hypotenuse.

Derive Pythagorean Identity Example

Derive Pythagorean Identity (Unit Circle) Example