sin(x)=2sin(x/2)cos(x/2), cos(x)=1-2sin²(x/2), so:
sin(x)+cos(x)=1 can be written 2sin(x/2)cos(x/2)+1-2sin²(x/2)=1.
Therefore, the 1s cancel: 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, so x=0, 2π, 4π, 6π,...
Or, cos(x/2)=sin(x/2), tan(x/2)=1, x/2=π/4, x=π/2, 3π/2, 5π/2,...