To derive the pythagorean identity the lengths of the adjacent and opposite side of the right triangle are defined in terms of the cosine and sine of the angle of the right-triangle. Then, the lengths are substituted into Pythagorean’s Theorem .

Lets start with the triangle formed by the unit circle, which visualizes all right triangles of hypotenuse one. Note, the adjacent side corresponds to the x-component of the right triangle and the opposite side corresponds to the y-component of the right triangle. The two components form the (x,y) point along the circumference of the circle.

Then, using the definitions of the trigonometric functions cosine and sine, we can substitute the variables which represent the adjacent side (x), the opposite side (y), and hypotenuse (1) of the right triangle into the equations.

After simplifying the equations, the adjacent side corresponds directly with the cosine function and the opposite side corresponds with the sine function for a given angle.

Next, recall the equation for Pythagorean’s Theorem which relates the squares of the sides together as shown below:

After substituting the corresponding variables to convert the theorem into the Cartesian Coordinate System we are left with a familiar equation, the equation of a circle.

Then, by substituting the corresponding sine and cosine function above, which we found to correspond to the x and y components of the triangle, we get Pythagorean’s identity.

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 cosine function returns the cosine of an angle provided in radians. In geometric terms, the function returns the ratio of the right-triangle'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 pythagorean theorem equates the square of the sides of a right triangle together.

The pythagorean identity relates the sides of the right triangle together using only the angle of the right triangle. The identity is derived using pythagorean's theorem and the properties of the unit circle.