Derivation of the First Pythagorean Identity using the trigonometric ratios


Mike Rosonet

Given the Pythagorean Theorem, that a = y, b = x, and c = r, and the equivalent trigonometric ratios, the following is true by substitution:

Through some algebraic manipulation, solving each equation for a and b, respectively, yields the new equations:

Now, substitute these new values for a and b into the Pythagorean Theorem and simplify:

Thus, by using the sides of the triangle in terms of the trigonometric ratios of the sides of the triangles, it is also possible to derive the preceding Pythagorean Identity.