$\cos \theta \left[ \begin{array} { c c } { \cos \theta } & { \sin \theta } \\ { - \sin \theta } & { \cos \theta } \end{array} \right] + \sin \theta \left[ \begin{array} { c c } { \sin \theta } & { - \cos \theta } \\ { \cos \theta } & { \sin \theta } \end{array} \right]$
$= \left[ \begin{array} { c c } { \cos ^ { 2 } \theta } & { \sin \theta \cos \theta } \\ { - \sin \theta \cos \theta } & { \cos ^ { 2 } \theta } \end{array} \right] + \left[ \begin{array} { c c } { \sin ^ { 2 } \theta } & { - \sin \theta \cos \theta } \\ { \sin \theta \cos \theta } & { \sin ^ { 2 } \theta } \end{array} \right]$
$= \left[ \begin{array} { c c } { \cos ^ { 2 } \theta + \sin ^ { 2 } \theta } & { \sin \theta \cos \theta - \sin \theta \cos \theta } \\ { - \sin \theta \cos \theta + \sin \theta \cos \theta } & { \cos ^ { 2 } \theta + \sin ^ { 2 } \theta } \end{array} \right]$
$ = \left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&{ 1} \end{array}} \right]\quad \left[ {\because {{\sin }^2}\theta + {{\cos }^2}\theta = 1} \right]$
= I = unit matrix.
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