Sam asked about this problem:
I’m going to show off because (a) I got this an about 2 minutes, and (b) I totally geeked out and spent 40 minutes making an animation for it.
It’s first worth noting that sine is asymmetric, and cosine is symmetric. Switching the sign of one of those ends up being just like spinning your vector around the other way. The numerator, then, represents the y component of two unit vectors added together, while the denominator represents the x component. You can then represent the problem as two vectors end to end, rotating in opposite directions. You will note from the animation below that the end point ends up drawing out a straight line. Since the slope of that line is constant, the relative x and y offsets must be proportional, and the ratio above is constant as well.