GIven: $\mathrm{A}=\left[\begin{array}{cc}\cos \mathrm{\theta }& \sin \mathrm{\theta }\\ -\sin \mathrm{\theta }& \cos \mathrm{\theta }\end{array}\right]$
Now we need to find A²
So,
${\mathrm{A}}^2=\left[\begin{array}{cc}\cos \mathrm{\theta }& \sin \mathrm{\theta }\\ -\sin \mathrm{\theta }& \cos \mathrm{\theta }\end{array}\right]\left[\begin{array}{cc}\cos \mathrm{\theta }& \sin \mathrm{\theta }\\ -\sin \mathrm{\theta }& \cos \mathrm{\theta }\end{array}\right]$

$\Rightarrow {\mathrm{A}}^2=\left[\begin{array}{cc}{\cos }^2\mathrm{\theta }-{\sin }^2\mathrm{\theta }& \cos \mathrm{\theta }\sin \mathrm{\theta }+\sin \mathrm{\theta }\cos \mathrm{\theta }\\ -\sin \mathrm{\theta }\cos \mathrm{\theta }-\sin \mathrm{\theta }\cos \mathrm{\theta }& -{\sin }^2\mathrm{\theta }+{\cos }^2\mathrm{\theta }\end{array}\right]$
$\Rightarrow {\mathrm{A}}^2=\left[\begin{array}{cc}{\cos }^2\mathrm{\theta }-{\sin }^2\mathrm{\theta }& 2\sin \mathrm{\theta }\cos \mathrm{\theta }\\ -2\sin \mathrm{\theta }\cos \mathrm{\theta }& {\cos }^2\mathrm{\theta }-{\sin }^2\mathrm{\theta }\end{array}\right]$
$\Rightarrow {\mathrm{A}}^2=\left[\begin{array}{cc}\cos 2\mathrm{\theta }& \sin 2\mathrm{\theta }\\ -\sin 2\mathrm{\theta }& \cos 2\mathrm{\theta }\end{array}\right]$