(v)
\[\begin{array}{*{35}{l}}
2\text{ }si{{n}^{2}}~A\text{ }+\text{ }co{{s}^{2}}~A \\
=\text{ }2\text{ }si{{n}^{2}}~A\text{ }+\text{ }{{\left( 1\text{ }-\text{ }si{{n}^{2}}~A \right)}^{2}} \\
=\text{ }2\text{ }si{{n}^{2}}~A+\text{ }1\text{ }+\text{ }si{{n}^{4}}~A\text{ }-\text{ }2\text{ }si{{n}^{2}}~A \\
=\text{ }1\text{ }+\text{ }si{{n}^{4}}~A\text{ }=\text{ }RHS \\
\end{array}\]
(vi)