Answer
$${\left( {2\sin x + \cos x} \right)^2} + {\left( {2\cos x - \sin x} \right)^2} = 5$$
Work Step by Step
$$\eqalign{
& {\left( {2\sin x + \cos x} \right)^2} + {\left( {2\cos x - \sin x} \right)^2} = 5 \cr
& {\text{We transform the more complicated left side to match the right side}}. \cr
& {\left( {2\sin x + \cos x} \right)^2} + {\left( {2\cos x - \sin x} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\, = 4{\sin ^2}x + 4\sin x\cos x + {\cos ^2}x + 4{\cos ^2}x - 4\sin x\cos x + {\sin ^2}x \cr
& \,\,\,\,\,\,\,\,\,\,\, = 5{\sin ^2}x + 5{\cos ^2}x \cr
& \,\,\,\,\,\,\,\,\,\,\, = 5\left( {{{\sin }^2}x + {{\cos }^2}x} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\, = 5 \cr
& {\text{Thus have verified that the given equation is an identity}} \cr} $$
