1-cosx+i*sinx
=2[sin(x/2)]^2+i*2sin(x/2)cos(x/2)
=2sin(x/2)[sin(x/2)+i*cos(x/2)]
=2sin(x/2)[cos(π/2-x/2)+i*sin(π/2-x/2)]
=2sin(x/2)*e^[i*(π/2-x/2)] .