Question

Prove F(x)=sin^2(x) and G(x)=cos^2(x) have the same derivatives. What can you say about the difference f-g? Explain using a consequence of the Mean Value theorem?

Answer 1

whoa

Answer 2

F + G = 1

F' + G' = 0

F' = - G'

**same** derivatives??

Answer 3

Another way of showing the claim is wrong:
\[\sin^2x=\frac{1-\cos2x}{2}~~\implies~~\frac{d}{dx}\sin^2x=\sin2x\\
\cos^2x=\frac{1+\cos2x}{2}~~\implies~~\frac{d}{dx}\cos^2x=-\sin2x\]