Minimum and maximum of $ \sin^2(\sin x) + \cos^2(\cos x) $ I want to find the maximum and minimum value of this expression:
$$ \sin^2(\sin x) + \cos^2(\cos x) $$

Your expression simplifies to 
$$1+\cos(2\cos x)-\cos (2\sin x).$$
We optimize of $1+\cos u-\cos v$ under the constraint $u^2+v^2=4$.
$\cos$ is an even function, so we can say that we optimize $1+\cos 2u-\cos(2\sqrt{1-u^2})$, $u\in [0,1]$, which should be doable.
