# Trigonometric Proof of the Pythagorean Theorem

Elisha Loomis, myself and no doubt many others believed and still believe that no trigonometric proof of the Pythagorean theorem is possible. This belief stems from the assumption that any such proof would rely on the most fundamental of trigonometric identities *sine* and *cosine* without a recourse to

I happily admit to being in the wrong.

Here are the two subtraction formulas that J. Zimba employs in his derivation of the Pythagorean theorem:

cos (α - β) = cos α cos β + sin α sin β,

sin (α - β) = sin α cos β - cos α sin β.

Note that the common definitions of sine and cosine as well as the proofs of the two identities always restrict α and β to the positive values less than 90°. For this reason, simply setting α = β in the first of these leads to *definition by convenience*. As such, it could not be reasonably used in a proof of such a fundamental result as the Pythagorean theorem. Instead, assume that *y* < *x* < 90°.

cos y | = cos (x - (x - y)) | |

= cos x cos(x - y) + sin x sin(x - y) | ||

= cos x (cos x cos y + sin x sin y) + sin x (sin x cos y - cos x sin y) | ||

= (cos²x + sin²x)cos y, |

which implies sin²*x* + cos²*x* = 1 since, by the definition, *y*

### References

- E. S. Loomis,
*The Pythagorean Proposition*, NCTM, 1968 - J. Zimba,
__On the Possibility of Trigonometric Proofs of the Pythagorean Theorem__,*Forum Geometricorum*, Volume 9 (2009) 275-278

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