I think you might mean:
sin2(2x)+(2sin(45+x)cos(45+x))2=1.
Note that sin(45)=cos(45)=1√2.
sin(45+x)=sin(45)cos(x)+cos(45)sin(x)=(1/√2)(cos(x)+sin(x));
cos(45+x)=cos(45)cos(x)-sin(45)sin(x)=(1/√2)(cos(x)-sin(x)).
2sin(45+x)cos(45+x)=2(1/√2)(1/√2)(cos2(x)-sin2(x))=cos2(x)-sin2(x)=cos(2x).
Therefore:
sin2(2x)+(2sin(45+x)cos(45+x))2=sin2(2x)+cos2(2x)=1