Consider an object experiencing uniform circular motion. What are the equations giving x and y as a function of time for such an object?

The equation for x(t) for our object experiencing uniform circular motion looks the same as the one we derived for the mass on the spring experiencing simple harmonic motion.

Clearly there's a parallel here - when A = R and the two ω's are equal, the two motions in the x-direction are identical (given appropriate initial conditions).