JEE Main & Advanced Mathematics Trigonometric Identities Question Bank Trigonometrical ratios of multiple and sub multiple angles

  • question_answer
    If \[\cos \theta =\frac{3}{5}\]and \[\cos \varphi =\frac{4}{5},\]where \[\theta \]and \[\varphi \]are positive acute angles, then \[\cos \frac{\theta -\varphi }{2}=\] [MP PET 1988]

    A) \[\frac{7}{\sqrt{2}}\]

    B) \[\frac{7}{5\sqrt{2}}\]

    C) \[\frac{7}{\sqrt{5}}\]

    D) \[\frac{7}{2\sqrt{5}}\]

    Correct Answer: B

    Solution :

    We have \[\cos \theta =\frac{3}{5}\]and\[\cos \varphi =\frac{4}{5}\]. Therefore \[\cos (\theta -\varphi )=\cos \theta \cos \varphi +\sin \theta \sin \varphi \]      \[=\frac{3}{5}.\frac{4}{5}+\frac{4}{5}.\frac{3}{5}=\frac{24}{25}\] But \[2{{\cos }^{2}}\left( \frac{\theta -\varphi }{2} \right)=1+\cos (\theta -\varphi )=1+\frac{24}{25}=\frac{49}{50}\] \\[{{\cos }^{2}}\left( \frac{\theta -\varphi }{2} \right)=\frac{49}{50}\].  Hence, \[\cos \left( \frac{\theta -\varphi }{2} \right)=\frac{7}{5\sqrt{2}}\].

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