Multiply through by cos(x): sin(x)+cos(x)=1.
sin(x)=2sin(x/2)cos(x/2); cos(x)=1-2sin2(x/2).
2sin(x/2)cos(x/2)+1-2sin2(x/2)=1,
2sin(x/2)cos(x/2)-2sin2(x/2)=0,
2sin(x/2)(cos(x/2)-sin(x/2))=0.
Therefore, sin(x/2)=0, x/2=πn, where n is an integer, x=2πn
Or, cos(x/2)=sin(x/2), tan(x/2)=1, x/2=π/4+πn, x=π/2+2πn.