The product of matrices $\mathrm{A}=\left[\begin{array}{cc}{\cos }^2\mathrm{\theta }& \cos \mathrm{\theta sin}\mathrm{\theta }\\ \cos \mathrm{\theta sin}\mathrm{\theta }& {\sin }^2\mathrm{\theta }\end{array}\right]$ and $\sin \mathrm{B}=\left[\begin{array}{cc}{\cos }^2\mathrm{\varphi }& \cos \mathrm{\varphi sin}\mathrm{\varphi }\\ \cos \mathrm{\varphi sin}\mathrm{\varphi }& {\sin }^2\mathrm{\varphi }\end{array}\right]$ is a null matrix $\mathrm{then} \theta -\varphi =$

$\mathrm{AB}=\left[\begin{array}{cc}{\cos }^2\mathrm{\theta }& \cos \mathrm{\theta sin}\mathrm{\theta }\\ \cos \mathrm{\theta sin}\mathrm{\theta }& {\sin }^2\mathrm{\theta }\end{array}\right]\left[\begin{array}{cc}{\cos }^2\mathrm{\varphi }& \cos \mathrm{\varphi sin}\mathrm{\varphi }\\ \cos \mathrm{\varphi sin}\mathrm{\varphi }& {\sin }^2\mathrm{\varphi }\end{array}\right]$

     $=\left[\begin{array}{l}{\cos }^2{\mathrm{\theta cos}}^2\mathrm{\varphi }+\cos \mathrm{\theta cos}\mathrm{\varphi sin}\mathrm{\theta sin}\mathrm{\varphi }\\ \cos \mathrm{\theta sin}{\mathrm{\theta cos}}^2\mathrm{\varphi }+{\sin }^2\mathrm{\theta cos}\mathrm{\varphi sin}\mathrm{\varphi }\end{array}\right.\left.\begin{array}{l}{\cos }^2\mathrm{\theta cos}\mathrm{\varphi sin}\mathrm{\varphi }+\cos \mathrm{\theta sin}{\mathrm{\theta sin}}^2\mathrm{\varphi }\\ \cos \mathrm{\theta cos}\mathrm{\varphi sin}\mathrm{\theta sin}\mathrm{\varphi }+{\sin }^2{\mathrm{\theta sin}}^2\mathrm{\varphi }\end{array}\right]$

$\begin{array}{c}=\left[\begin{array}{ll}\cos \mathrm{\theta cos}\mathrm{\varphi }(\cos (\mathrm{\theta }-\mathrm{\varphi }\left)\right)& \cos \mathrm{\theta sin}\mathrm{\varphi }(\cos (\mathrm{\theta }-\mathrm{\varphi }\left)\right)\\ \sin \mathrm{\theta cos}\mathrm{\varphi }(\cos (\mathrm{\theta }-\mathrm{\varphi }\left)\right)& \sin \mathrm{\theta sin}\mathrm{\varphi }(\cos (\mathrm{\theta }-\mathrm{\varphi }\left)\right)\end{array}\right]\\ =(\cos (\mathrm{\theta }-\mathrm{\varphi }\left)\right)\left[\begin{array}{cc}\cos \mathrm{\theta cos}\mathrm{\varphi }& \cos \mathrm{\theta sin}\mathrm{\varphi }\\ \sin \mathrm{\theta cos}\mathrm{\varphi }& \sin \mathrm{\theta sin}\mathrm{\varphi }\end{array}\right]\end{array}$

$\text{ Now }\mathrm{AB}=\mathrm{O}\Rightarrow \cos (\mathrm{\theta }-\mathrm{\varphi })=0\Rightarrow \mathrm{\theta }-\mathrm{\varphi }=(2\mathrm{n}+1)\frac{\mathrm{\pi }}{2},\mathrm{n}\in \mathrm{Z}$