Let

$A=\frac{\sqrt{2}\cos 45°+\cos 56°+\cos 58°-\cos 66°}{\sqrt{2}\cos 28°\cos 29°\sin 33°}$

$\Rightarrow A=\frac{\left(1-\cos 66°\right)+\left(\cos 56°+\cos 58°\right)}{\sqrt{2}\cos 28°\cos 29°\sin \left(90-57\right)°}$

$\left[\because \cos 45°=\frac{1}{\sqrt{2}}\right]$

$\Rightarrow A=\frac{\left(1-1+2{\sin }^{2}33°\right)+2\cos 57°\cos 1°}{\sqrt{2}\cos 28°\cos 29°\sin \left(90-57\right)°}$

$\left[\begin{array}{c}\because \cos C+\cos D=2\cos \left(\frac{C+D}{2}\right)\cos \left(\frac{C-D}{2}\right)\\ \cos 2A=1-2{\sin }^{2}A\end{array}\right]$

$\Rightarrow A=\frac{2{\cos }^{2}57°+2\cos 57°\cos 1°}{\sqrt{2}\cos 28°\cos 29°\cos 57°}$

$\Rightarrow A=\frac{\sqrt{2}\left(\cos 57°+\cos 1°\right)}{\left(\cos 28°\cos 29°\right)}$

$\Rightarrow A=\frac{2\sqrt{2}\left(\cos 57°+\cos 1°\right)}{\left(2\cos 28°\cos 29°\right)}$

$\Rightarrow A=\frac{2\sqrt{2}\left(\cos 57°+\cos 1°\right)}{\cos 57°+\cos 1°}$

$\left[\because 2\cos A\cos B=\cos \left(A+B\right)+\cos \left(A-B\right)\right]$

$\Rightarrow A=2\sqrt{2}$