To derive the Pythagorean identity the lengths of the adjacent and opposite sides 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.
Let’s start with the triangle formed by the unit circle, which visualizes all right triangles of hypotenuse one. Note, that 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.