(b) Let $u = \cos \theta \left\{ {\sin \theta + \sqrt {{{\sin }^2}\theta + {{\sin }^2}\alpha } } \right\}$

==> ${(u – \sin \theta \cos \theta )^2} = {\cos ^2}\theta ({\sin ^2}\theta + {\sin ^2}\alpha )$ 

==> ${u^2}{\tan ^2}\theta – 2u\tan \theta + {u^2} – {\sin ^2}\alpha = 0$ 

Since tan $\theta $ is real, therefore 

==> $4{u^2} – 4{u^2}({u^2} – {\sin ^2}\alpha ) \ge 0$

$ \Rightarrow {u^2} – (1 + {\sin ^2}\alpha ) \le 0$ 

==> $|u|\, \le \sqrt {1 + {{\sin }^2}\alpha } $.