Maths help complex numbers

[bravegirl123]

The variable complex number z is given by z = 2cosθ + i(1 − 2 sinθ). Prove that the real part of 1/(z + 2 − i)
is constant for −π < θ < π.

I simplified the answer to (1/4)((cosθ +isinθ)/(1+cosθ)). Marking scheme says that real part equals 1/4. I thought that whatever is not multiplied with i is wholly the real part. Please explain why (1/4)(cosθ)/(1+cosθ) is not correct.

 

[aseelz]

The variable complex number z is given by z = 2cosθ + i(1 − 2 sinθ). Prove that the real part of 1/(z + 2 − i)
is constant for −π < θ < π.

I simplified the answer to (1/4)((cosθ +isinθ)/(1+cosθ)). Marking scheme says that real part equals 1/4. I thought that whatever is not multiplied with i is wholly the real part. Please explain why (1/4)(cosθ)/(1+cosθ) is not correct.