The maximum value of ${\cos }^2\left({45}^{\circ }+\mathrm{x}\right)+(\sin \mathrm{x}-\cos \mathrm{x}{)}^2$ is

$\begin{array}{l}{\cos }^2\left({45}^{\circ }+\mathrm{x}\right)+(\sin \mathrm{x}-\cos \mathrm{x}{)}^2\\ ={\left[\frac{\cos \mathrm{x}}{\sqrt{2}}-\frac{\sin \mathrm{x}}{\sqrt{2}}\right]}^2+(\sin \mathrm{x}-\cos \mathrm{x}{)}^2\\ =\frac{3}{2}(1-\sin 2\mathrm{x})\end{array}$

Hence, Maximum value $=\frac{3}{2}(1-(-1\left)\right)=3$