For any real $\theta$ , find the maximum value of  ${\cos }^2\left(\cos \theta \right)+{\sin }^2\left(\sin \theta \right)-{\sin }^{2}1$

The maximum value of  ${\cos }^2\left(\cos \theta \right)$  is 1 and that of  ${\sin }^2\left(\sin \theta \right)$ is ${\sin }^{2}1$ , both exists for  $\theta =\frac{\pi }{2}.$  Hence maximum value is    $1+{\sin }^{2}1$