sin(45+A)=sin45cosA+cos45sinA=(cosA+sinA)/√2; similarly cos(45+B)=(cosB-sinB)/√2.
Let A = 2sin(45+A)cos(45+B) = (cosA+sinA)(cosB-sinB) = cosAcosB-cosAsinB+sinAcosB-sinAsinB.
cos(A+B)=cosAcosB-sinAsinB; sin(A-B)=sinAcosB-cosAsinB.
So A=(cosAcosB-sinAsinB)+(sinAcosB-cosAsinB)=cos(A+B)+sin(A-B) QED.