Question

If $z=\frac{1}{1-\mathrm{cos\theta }-i\mathrm{sin\theta }}$, then Re (z) =
(a) 0
(b) $\frac{1}{2}$
(c) $\cot \frac{\mathrm{\theta }}{2}$
(d) $\frac{1}{2}\cot \frac{\mathrm{\theta }}{2}$

Answer 1

(b) $\frac{1}{2}$
$$\begin{gathered} z=\frac{1}{1-\cos \theta -i\sin \theta } \\ z=\frac{1}{1-\cos \theta -i\sin \theta }\times \frac{1-\cos \theta +i\sin \theta }{1-\cos \theta +i\sin \theta } \\ \Rightarrow z=\frac{1-\cos \theta +i\sin \theta }{{\left(1-\cos \theta \right)}^2-{\left(i\sin \theta \right)}^2} \\ \Rightarrow z=\frac{1-\cos \theta +i\sin \theta }{1+{\cos }^2\theta -2\cos \theta +{\sin }^2\theta } \\ \Rightarrow z={\displaystyle \frac{1-\cos \theta +i\sin \theta }{1+1-2\cos \theta }} \\ \Rightarrow z=\frac{1-\cos \theta +i\sin \theta }{2(1-\cos \theta )} \\ \Rightarrow \mathrm{R}\mathrm{e}(z)=\frac{\left(1-\cos \theta \right)}{2\left(1-\cos \theta \right)}=\frac{1}{2} \end{gathered}$$