$\text{ For any real }\theta \text{ the max value of }{\cos }^2\left(\cos \theta \right)+{\sin }^2\left(\sin \theta \right)=$

$\begin{array}{l}Let f\left(\theta \right)={\cos }^2\left(\cos \theta \right)+{\sin }^2\left(\sin \theta \right)\\ We have -1\le \cos \theta \le 1 \text{ and }-1\le \sin \theta \le 1\\ \therefore {\cos }^{2}1\le {\cos }^2\left(\cos \theta \right)\le 1\text{ and }0\le {\sin }^2\left(\sin \theta \right)\le {\sin }^{2}1\end{array}$

$\text{ Max value of }f\left(\theta \right)=1+{\sin }^{2}1$