Sin(x)^2 + Cos(x)^2 = 1 for all rational numbers x. I discovered this while punching random characters on my calculator.
A natural generalization is to generalise this identity to the Gaussian Integers. It would be interesting to see whether there only finitely many solutions to the equation when constrained to the Gaussian Integers and i /= 0 or the Imaginary Numbers. It would also be interesting to consider whether there are infinitely many solutions if 1 was replaced with 0. Here sin(x) is expressed in radians.
A elementary proof of this identity for rational number cases can be found in the link below.
https://en.wikipedia.org/wiki/Pythagorean_trigonometric_identity
My Mathematics work and other extraneous diversions 