If x+iy = 1/1+ cos theta + i sin theta , find x square

I presume the expression is:

x + iy = 1/[1 + cos θ + i sin θ)

The first step is to get this into standard form where the denominator is purely real. Multiply top and bottom by the complex conjugate.

x + iy = (1 + cos θ - i sin θ)/ [ (1+cos θ)^2 + sin^2 θ]

expand the denominator and use the fact that cos^2 + sin^2 = 1 to get
x + iy = (1 + cos θ - i sin θ)/[2 (1 + cos θ)]

x is the real part which is

x = (1 + cos θ)/[2 (1 + cos θ)] = 1/2
x^2 = 1/4

plz approve!