The “Impossible” Trigonometric Proof of the Pythagorean Theorem Explained

Why a Trigonometric Proof of the Pythagorean Theorem Was Considered Impossible

The Pythagorean theorem needs no introduction — the square of the hypotenuse equals the sum of the squares of the two legs. Mathematicians have found over 400 ways to prove it, ranging from geometric to algebraic approaches. Yet for well over a century, not a single trigonometric proof existed. Why?

In 1907, mathematician Elisha Scott Loomis declared that proving the Pythagorean theorem using trigonometry was impossible. His argument: trigonometry itself exists because of the Pythagorean theorem. In other words, all trigonometric identities are ultimately derived from the theorem — so any attempt to use them as proof creates circular reasoning. Loomis was believed, and for decades the mathematical community treated a trigonometric proof as an unsolvable problem.

Two High School Students Who Cracked the Pythagorean Theorem

In December 2022, that “impossible” challenge appeared as a bonus problem at a high school math competition in Louisiana. Two seniors, Ne’Kiya Jackson and Calcea Johnson, did what generations of mathematicians had dismissed as undoable — they constructed a valid trigonometric proof of the Pythagorean theorem.

The Proof: Using the Law of Sines to Avoid Circular Reasoning

Jackson and Johnson’s key insight was to use the Law of Sines — a rule for relating the sides and angles of a triangle that does not depend on the Pythagorean theorem. Rather than relying on standard trigonometric identities, they divided the original right triangle into a series of smaller triangles.

By applying the Law of Sines to each of these smaller triangles, they expressed the sides in terms of angles, carefully sidestepping the circular reasoning that had trapped every previous attempt. This approach allowed them to derive the relationship for the hypotenuse without assuming it to be true in advance.

Why This Trigonometric Proof of the Pythagorean Theorem Matters

Their work demonstrates that “impossible” is rarely a permanent verdict — even in a field as thoroughly studied as classical mathematics. A theorem proven hundreds of times over more than two millennia still had a genuinely new angle waiting to be discovered.

All it took was a fresh perspective.

The original paper is available here.