Hello everyone, I am having some troubles with the following problem:

Find a derivative (Fréchet derivative) of the operator

$\displaystyle F:C[0,\pi]\to C[0,\pi], F(u)=\sin u(x),$

at the point $\displaystyle u_0(x) = \cos(x).$

There is an answer to this problem in the book: $\displaystyle F'(u_0)=\cos\sin x$, but I can't get it: I tried using mean value theorem, Taylor series and different trigonometry formulas to find the linear part of the increment, but I still don't understand how can we get an answer $\displaystyle \cos\sin x$ here. I also don't thinks its necessary to find Gateaux derivative here, which is equal to Frechet derivative in this case. There is also a similar problem with $\displaystyle F(u)=\cos u(x), u_0(x) = \sin(x)$ with an answer $\displaystyle \sin\cos x$ in this book, so I think this is not any mistake or something. Can anyone please help with this?

Thanks in advance.