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We also have
We assume the knowledge
cartesian(x) *C cartesian(y) = (x0cos(x1) + x0sin(x1)i) *C (y0cos(y1) + y0sin(y1)i) = (x0y0cos(x1)cos(y1) - x0y0sin(x1)sin(y1)) + (x0y0cos(x1)sin(y1) + x0y0sin(x1)cos(y1))i = x0y0(cos(x1)cos(y1) - sin(x1)sin(y1)) + x0y0(cos(x1)sin(y1) + x0y0sin(x1)cos(y1))i
as granted and are therefore done.
cos(x1+y1) = cos(x1)cos(y1) - sin(x1)sin(y1) sin(x1+y1) = cos(x1)sin(y1) + sin(x1)cos(y1)