In finding the derivative of g(x) = (cosx)^3

we get, g'(x) = 3(cosx)^2(-sinx)

but I don't understand how this can be simplified to

g'(x) = -3cos^2xsinx Why is is permissible to move the power of 2 to be immediately after cos instead of
after cosx?

It's simply a notational thing that's commonly used with trigonometric functions. Cos^2 is understood to mean the same as (cos x)^2, so as not to be confused with cos (x^2).