find the maximum value of
(cos(cos x))^2 +(sin(sin x))^2
The function
$\displaystyle f(x)= (\;\cos (\cos x)\^2+(\;\sin (\sin x)\^2$
is even and periodic with period $\displaystyle \pi$ so, we only need to find the maximum on $\displaystyle [0,\pi/2]$ .
The values of $\displaystyle f$ at the endpoints of $\displaystyle [0,\pi/2]$ are:
$\displaystyle f(0)=\cos^2 1<\sin^2 1=f(\pi/2)$
Prove that there are no singular points in $\displaystyle (0,\pi/2)$ so, the absolute maximun of $\displaystyle f$ is $\displaystyle \sin^2 1$ and the absolute minimum $\displaystyle \cos^2 1$ .