\sin{t} + i (1-\cos{t})=
 \sin{t} + i*2\sin^2{t\over 2} =
2\sin{t\over 2}\cos{t\over 2} + i*2\sin^2{t\over 2} =
2\sin{t\over 2}(\cos{t\over 2}+i\sin{t\over 2})