(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)