The variable complex number z is given by z=1+cos 2θ+isin2θ,where θ takes all values in the interval −1/2π< θ< 1/2π

(i) show that the modulus of z is 2 cos θ and the argument of z is θ
(ii) prove that the real part of 1/z is constant.

The given complex number is
 z = 1 + cos(2θ) + i sin(2θ), for -1/2π < θ < 1/2π

Part (i)
Let V = the modulus of z.
Then
V² = [1 + cos(2θ)]² + sin²(2θ)
     = 1 + 2 cos(2θ) + cos²2θ + sin²2θ

Because sin²x + cos²x = 1, therefore
V² = 2(1 + cos2θ)

Because cos(2x) = 2 cos²x - 1, therefore
V² = 2(1 + 2cos²θ - 1) = 4 cos²θ
Because  -1/2π < θ < 1/2π,
V = 2 cosθ                          PROVEN

Part ii.
1/z = 1/[1 + cos2θ + i sin 2θ]


The denominator is


Therefore


The real part of 1/ = 1/  (constant).