JEE Main & Advanced Sample Paper JEE Main Sample Paper-47

  • 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

    Correct Answer: B

    Solution :

     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 \[=1+{{\sin }^{2}}1\]

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