This can be written:
sin(2x)+2sin2(45-x)=1 since sin2(x)+cos2(x)≡1.
sin(45-x)≡sin(45)cos(x)-cos(45)sin(x)=(cos(x)-sin(x))/√2, because sin(45)=cos(45)=1/√2.
sin2(45-x)≡½(cos2(x)-2sin(x)cos(x)+sin2(x))≡½(1-sin(2x)),
2sin2(45-x)≡1-sin(2x).
Therefore: sin(2x)+2sin2(45-x)≡sin(2x)+1-sin(2x)≡1≡sin2(x)+cos2(x) QED