Answer
$$(2\sin x+\cos x)^2+(2\cos x-\sin x)^2=5$$
We simplify the left side and find that the above expression is an identity.
Work Step by Step
$$(2\sin x+\cos x)^2+(2\cos x-\sin x)^2=5$$
We start dealing with the left side.
$$A=(2\sin x+\cos x)^2+(2\cos x-\sin x)^2$$
$$A=(4\sin^2x+4\sin x\cos x+\cos^2x)+(4\cos^2 x-4\sin x\cos x+\sin^2 x)$$
$$A=5\sin^2x+5\cos^2x$$
$$A=5(\sin^2x+\cos^2x)$$
$$A=5\times1=5$$ (since $\sin^2x+\cos^2x=1$)
So we have proved that the given expression is an identity.