$z = \frac{{1 – i\sin \theta }}{{1 + i\cos \theta }} \times \frac{{1 – i\cos \theta }}{{1 – i\cos \theta }}$

$ = \frac{{1 – i(\cos \theta + \sin \theta ) + {i^2}\sin \theta \cos \theta }}{{1 – {i^2}{{\cos }^2}\theta }}$

$ = \frac{{1 – \sin \theta \cos \theta – i(\cos \theta + \sin \theta )}}{{1 + {{\cos }^2}\theta }}$

$\because {\mathop{\rm Im}\nolimits} (z) = 0,\cos \theta + \sin \theta = 0$

$\Rightarrow \tan \theta = -1 $

$\Rightarrow \theta = n\pi – \frac {\pi}{4}$,$n\in I$