I don't know what would constitute an actual answer to this question. Merely restating various points about Taylor series doesn't seem like it says anything since the question seems to be why the cosine approximation, term for term, is worse than the sine approximation (near 0). This is not intuitive, since $\sin$ and $\cos$ are just shifted versions of each other—shouldn't their approximations be as good?
For comparison, consider one of the popular interpolation methods: cubic splines. We like cubic splines because 1) they hit every point 2) the slopes match at the points 3) the curvature matches at the points.
What would it mean to constitute a good approximation? We often like to approximate with low-order polynomials (e.g. "everything is approximately a line if you zoom in enough", Simpson's rule uses a parabola to estimate, etc.). Since the derivative of a line is a constant, this kind of approximation should be alright if the derivative isn't changing very much (the derivative of a line doesn't change at all). This is by analogy to the point (3) of cubic splines. Since the rate of change of the derivative of $\sin(x)$ is (ignoring the sign) just $\sin(x)$ and at 0 the sine is 0, this means our small approximation should be alright: it's not a line, but close to 0 it sure acts like one . If you compare this to cosine the situation is reversed and we are at the maximum rate of change of the derivative, so our small approximation should have more error. It's not a line, and it isn't acting like one, either.