use derivative to prove identity?

Question

sin^2x+cos^2x=1

Answer 1

Let f(x) = (sinx)^2 + (cosx)^2. Then if the derivate is 0 for all x, that means f(x) is constant.

The derivative is 2(sinx)(cosx) + 2(cosx)(-sinx) =0. Hence, f(x) is a constant function.

To see that f(x) = 1, just note that f(0) = 0^2 + 1^2 = 1.