(i) The complex number $w$ can be expressed as $\cos \theta+i \sin \theta$.

(a) Show that $w+\frac{1}{-}$ is a real number.

[2]

(b) Show that $\frac{w-1}{w+1}$ can be expressed as $k \tan \frac{1}{2} \theta$, where $k$ is a complex number to be found.

[4]

(ii) The complex number $z$ has modulus 1 . Find the modulus of the complex number $\frac{z-3 \mathrm{i}}{1+3 \mathrm{i} z}$.

[5]