For any real value of $\theta$, the maximum value of  ${\cos }^2\left(\cos \theta \right)+{\sin }^2\left(\sin \theta \right)$  is

$-1\le \cos \theta \le 1$

$=\cos 1\le \cos \left(\cos \theta \right)\le 1$

$={\cos }^{2}1\le {\cos }^2\left(\cos \theta \right)\le 1\to \left(1\right)$

$=-1\le \sin \theta \le 1$

$=-\sin 1\le \sin \left(\sin \theta \right)\le \sin 1$

$=0\le {\sin }^2\left(\sin \theta \right)\le {\sin }^{2}1\to \left(2\right)$

From (1) and (2),

$1+{\sin }^{2}1$