For this proof, it must be clear to you that [limx->0 (cos(x) - 1)/x = 0] and that [limx->0 sin(x)/x = 1]. You must also know how to expand cos(a + b), if you don't, then click here. By definition, the derivative of cos(x) would be:
If we expand cos(x + h):
If we keep simplifying it further:
We know that [limh->0 (cos(h) - 1)/h = 0], [limh->0 sin(x) = sin(x)] and [limx->0 sin(x)/x = 1], so:
Hence we have proved that the derivative of cos(x) is -sin(x).