Prove the identities:

1. $$(\sin x+\cos x{)}^2=1+\sin (2x)$$
2. $${\cos }^{4}x-{\sin }^{4}x=\cos (2x)$$
3. $$\frac{2\cos (2\theta )-1}{1+2\sin \theta }=1-2\sin \theta$$