• # question_answer If $\left| \cos \,\theta \,\left\{ \sin \theta +\sqrt{{{\sin }^{2}}\theta +{{\sin }^{2}}\alpha } \right\}\, \right|\,\le k,$ then the value of k is A) $\sqrt{1+{{\cos }^{2}}\alpha }$ B) $\sqrt{1+{{\sin }^{2}}\alpha }$ C) $\sqrt{2+{{\sin }^{2}}\alpha }$ D) $\sqrt{2+{{\cos }^{2}}\alpha }$

Let $u=\cos \theta \left\{ \sin \theta +\sqrt{{{\sin }^{2}}\theta +{{\sin }^{2}}\alpha } \right\}$ Þ ${{(u-\sin \theta \cos \theta )}^{2}}={{\cos }^{2}}\theta ({{\sin }^{2}}\theta +{{\sin }^{2}}\alpha )$ Þ ${{u}^{2}}{{\tan }^{2}}\theta -2u\tan \theta +{{u}^{2}}-{{\sin }^{2}}\alpha =0$ Since tan $\theta$is real, therefore Þ $4{{u}^{2}}-4{{u}^{2}}({{u}^{2}}-{{\sin }^{2}}\alpha )\ge 0$ $\Rightarrow {{u}^{2}}-(1+{{\sin }^{2}}\alpha )\le 0$ Þ $|u|\,\le \sqrt{1+{{\sin }^{2}}\alpha }$.