The basic insight is that sin is -1 times its second derivative, just as the second power of i is -1. So in both cases, you’ll have a pattern that negates with every two steps (and thus repeats with every four steps).
A) What about the fact that both patterns sync up, both starting with ++ on the 0th and 1st steps?
Well, if you have any sign pattern that negates with every two steps, it will go …-+±-+±-+±-… starting from some point.
If we’re dealing with powers of nicely named constant, the 0th power will always be +1, and the 1st power will be the nicely named constant, and so, by your convention to give it a nice name, will automatically also be considered +. This explains why the powers of i start ++. (Note that i isn’t actually positive, as such; it’s just, well, a positive multiple of i)
Similarly, if we’re dealing with function’s derivatives, the +s in our sign series will correspond to nicely named functions and the -s will correspond to their negations. There will be two nicely named functions (corresponding to the two distinct +s in each period of the series), one whose derivatives produce a series starting ++ and the other a series starting ±. In this case, you chose sine to start with (++) rather than cosine (±); you chose the one that would make it line up with the powers of i, rather than the one which wouldn’t. That’s all.
B) Why do we have this connection of negating every two steps in the first place? Why should i^2 = -1 and why should sin’’ = -sin?
As for why i^2 should equal -1, the answer is natural: “That’s what i is. That’s its defining property. The thing we mean by i is ‘some constant whose powers negate every two steps’”. No mystery there.
But actually, the same could be said of sin’’ = -sin. In some sense, this is what trigonometry is all about: the study of the differential equation f’’ = -f. We should have that sin’’ = -sin because that’s the defining property of sin (and cosine and rotation more generally) is. Rotation is the operation which is proportional to its negated second derivative.
Granted, this is not how most people originally start thinking about trigonometry, thanks to a wonderful quirk of our physical universe which both makes “geometry” instinctually intuitive to us, yet obscures its connections to other abstract mathematics. This differential equation is so baked into the world we live in that we don’t even recognize it anymore; we might instead start from a different framework (one that seems very basic to us because it connects very closely to our everyday physical experience), with a different definition of sine inside that framework, and then prove the differential equation from that framework. But there’s no reason we have to go about things that way. If we tried to explain rotation, Euclidean geometry, etc., to incorporeal, non-physical aliens, at some point, we would have no better motivation for our axioms than that they were random axioms we were interested in studying. Our alien friend would likely see the differential equation f’’ = -f as much more natural an object of study, then, than whatever complicated system of “geometric” axioms we tried to motivate it by.
So, to make the story short by repeating its introduction: the derivatives of the basic trig functions negate every two steps because that’s the defining property of the basic trig functions. We happen to be particularly interested in functions with that property because they play a pivotal rule in the arbitrary physical rules of our universe, but we might just as well be interested in them simply because they have the nice property of having their derivatives negate every two steps.