Use Distributive Property of FOIL method to obtain:

\(\sin^{2}x+2\sin x\cos x+\cos^{2}x\)

Use the identity: \(\sin^{2}x+\cos^{2}x=1\)

\(=1+2\sin x\cos x\)

Use the double-angle identity: \(\sin^{2}x=2\sin x\cos x\)

\(=1+\sin2x\)

\(\sin^{2}x+2\sin x\cos x+\cos^{2}x\)

Use the identity: \(\sin^{2}x+\cos^{2}x=1\)

\(=1+2\sin x\cos x\)

Use the double-angle identity: \(\sin^{2}x=2\sin x\cos x\)

\(=1+\sin2x\)