The value of ${\cos }^2\mathrm{A}{\left(3-4{\cos }^2\mathrm{A}\right)}^2+{\sin }^2\mathrm{A}{\left(3-4{\sin }^2\mathrm{A}\right)}^2$ is equal to

 

$\begin{array}{l}{\cos }^2\mathrm{A}{\left(3-4{\cos }^2\mathrm{A}\right)}^2+{\sin }^2\mathrm{A}{\left(3-4{\sin }^2\mathrm{A}\right)}^2\\ ={\left(3\cos \mathrm{A}-4{\cos }^3\mathrm{A}\right)}^2+{\left(3\sin \mathrm{A}-4{\sin }^3\mathrm{A}\right)}^2\\ =(-\cos 3\mathrm{A}{)}^2+(\sin 3\mathrm{A}{)}^2=1\end{array}$