21M.2.AHL.TZ2.8

$\frac{1+z}{1-z}=\frac{1+\cos \theta +\text{i} \sin \theta }{1-\cos \theta -\text{i} \sin \theta }$

attempt to use the complex conjugate of their denominator           M1

$=\frac{\left(1+\cos \theta +\text{i} \sin \theta \right)\left(1-\cos \theta +\text{i} \sin \theta \right)}{\left(1-\cos \theta -\text{i} \sin \theta \right)\left(1-\cos \theta +\text{i} \sin \theta \right)}$            A1

$\text{Re}\left(\frac{1+z}{1-z}\right)=\frac{1-{\cos }^2 \theta -{\sin }^2 \theta }{{\left(1-\cos \theta \right)}^2+{\sin }^2 \theta } \left(=\frac{1-{\cos }^2 \theta -{\sin }^2 \theta }{2-2 \cos \theta }\right)$          M1A1

Note: Award M1 for expanding the numerator and A1 for a correct numerator. Condone either an incorrect denominator or the absence of a denominator.

using ${\cos }^2 \theta +{\sin }^2 \theta =1$ to simplify the numerator           (M1)

$\text{Re}\left(\frac{1+z}{1-z}\right)=0$            AG

 

[5 marks]

[N/A]