$\frac{\mathit{s}\mathit{i}{\mathit{n}}^{\mathit{2}}\mathit{3}\mathit{A}}{\mathit{s}\mathit{i}{\mathit{n}}^{\mathit{2}}\mathit{A}}\mathit{-}\frac{\mathit{c}\mathit{o}{\mathit{s}}^{\mathit{2}}\mathit{A}}{\mathit{c}\mathit{o}{\mathit{s}}^{\mathit{2}}\mathit{A}}\mathit{=}$

$\begin{array}{rcl} & & \frac{{\sin }^{2}3\mathrm{A}}{{\sin }^2\mathrm{A}}-\frac{{\cos }^{2}3\mathrm{A}}{{\cos }^2\mathrm{A}}\\ & =& \frac{{\sin }^{2}3\mathrm{A} {\cos }^2\mathrm{A}-{\cos }^{2}3\mathrm{A} {\sin }^2\mathrm{A}}{{\sin }^2\mathrm{A} {\cos }^2\mathrm{A}}\\ & =& \frac{(\sin 3\mathrm{A} \cos \mathrm{A}-\cos 3\mathrm{A} \sin \mathrm{A}) (\sin 3\mathrm{A} \cos \mathrm{A}+\cos 3\mathrm{A} \sin \mathrm{A})}{{\sin }^2\mathrm{A} {\cos }^2\mathrm{A}}\\ & =& \frac{\sin 2\mathrm{A} \sin 4\mathrm{A}}{{\sin }^2\mathrm{A} {\cos }^2\mathrm{A}}\\ & =& \frac{2\sin \mathrm{A} \cos \mathrm{A} 2\sin 2\mathrm{A} \cos 2\mathrm{A}}{{\sin }^2\mathrm{A} {\cos }^2\mathrm{A}}\\ & =& \frac{4\mathrm{sinA} \cos \mathrm{A} 2\sin \mathrm{A} \cos \mathrm{A} \cos 2\mathrm{A}}{{\sin }^2\mathrm{A} {\cos }^2\mathrm{A}}\\ & =& 8\cos 2\mathrm{A}\end{array}$