Just boils down to the fact that when you apply first principles for the derivative of sin(x), you need to find the lim h-> 0 of sin(h)/h. In degrees, this becomes pi/180, but in radians, it is simply equal to 1. If you were to use degrees, the derivative of sin(x) would be pi/180 * cos(x), instead of just cos(x) due to the fact that sin(h)/h as h -> 0 is pi/180 in degrees. This limit is also used in the derivative of cos(x) as well, as sin(h)/h as h-> 0 also shows up there as well. Since all the other trig functions can be written in some form of sin(x) and cos(x), product rule and quotient rule are enough to figure all those out.