The Pythagorean Identity relates the sides of the right triangle together using only the angle of the right triangle. The identity is part of the collection of Trigonometric Identities and can be derived using pythagorean’s theorem and the properties of the unit circle^{[1]}. Geometrically, the components of the equation can be visualized by the right triangle below.
The pythagorean identity can be derived using the trigonometric functions cosine and sine and the right triangle with a hypotenuse of one. This is triangle is best demonstrated with the unit circle, which visualizes all right triangles of hypotenuse one.
Using the definitions of sine and cosine, we can substitute the variables which represent the adjacent, opposite, and hypotenuse sides of the right triangle into the equations. In this case, the adjacent side corresponds
When we substitute x in for the adjacent side, y for the opposite side, and 1 for the hypotenuse, we can see the adjacent side corresponds directly with the cosine function and the opposite side corresponds with the sine function. Finally, to finish deriving the pythagorean identity, we can combine these facts with Pythagorean’s Theorem, which relates the squares of the sides together.
First, convert the theorem into the cartesian coordinate system. This should feel very familiar as it is the equation of a circle.
Then, substitute the corresponding functions above into the equation to get Pythagorean’s 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.
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.
The trigonometric identites are a set of equations derived from the properties of the right triangle and the circle.
The cosine function returns the cosine of an angle provided in radians. In geometric terms, the function returns the ratio of the righttriangle's adjacent side over its hypotenuse.
The sine function returns the sine of a number provided in radians. In geometric terms, the function returns the vertical component of the point formed by the angle on the unit circle.
The unit circle is a circle of radius one placed at the origin of the coordinate system. This article discusses how the unit circle represents the output of the trigonometric functions for all real numbers.
The pythagorean theorem equates the square of the sides of a right triangle together.

Derive Pythagorean Identity (Unit Circle)Wumbo (internal)