• # question_answer For any real $\theta ,$ the maximum value of ${{\cos }^{2}}\,(\cos \,\theta )+{{\sin }^{2}}(\sin \theta )$ is A)  1                                 B)  $1+{{\sin }^{2}}1$ C)  $1+{{\cos }^{2}}1$                      D)  None of these

Let $f(\theta )=co{{s}^{2}}\,(\cos \,\theta )+{{\sin }^{2}}(\sin \,\theta )$ $\because$     $-1\le \,\cos \,\theta \le 1$ and $-1\le \sin \,\theta \le 1$ $\therefore$    $\cos \,1\le \,\cos \,(\cos \theta )\,\le 1$ and      $-\sin 1\le sin\,(sin\,\theta )\,\le \,sin\,1$ $\therefore$    ${{\cos }^{2}}1\le \,{{\cos }^{2}}\,(\cos \,\theta )\,\le 1$ and      $0\le {{\sin }^{2}}(\sin \,\theta )\,\le {{\sin }^{2}}1$ $\therefore$ Maximum value $=1+{{\sin }^{2}}1$