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

  • question_answer
    If \[\sin \beta \]is the geometric mean between \[\sin \alpha \]and \[\cos \alpha ,\]then \[\cos 2\beta \]is equal to

    A) \[2{{\sin }^{2}}\left( \frac{\pi }{4}-\alpha  \right)\]

    B) \[2{{\cos }^{2}}\left( \frac{\pi }{4}-\alpha  \right)\]

    C) \[2{{\cos }^{2}}\left( \frac{\pi }{4}+\alpha  \right)\]

    D) \[2{{\sin }^{2}}\left( \frac{\pi }{4}+\alpha  \right)\]

    Correct Answer: C

    Solution :

    Since \[\sin \beta \] is G.M. between \[\sin \alpha \]and \[\cos \alpha \]. \[\therefore \,\,{{\sin }^{2}}\beta =\sin \alpha \cos \alpha \] Now \[\cos 2\beta =1-2{{\sin }^{2}}\beta =1-2\sin \alpha \cos \alpha \] \[={{(\cos \alpha -\sin \alpha )}^{2}}=2\,{{\left( \frac{1}{\sqrt{2}}\cos \alpha -\frac{1}{\sqrt{2}}\sin \alpha  \right)}^{2}}\] \[=2{{\sin }^{2}}\left( \frac{\pi }{4}-\alpha  \right)\], which is given in (a). Also \[\cos 2\beta =2{{\cos }^{2}}\left\{ \frac{\pi }{2}-\left( \frac{\pi }{4}-\alpha  \right) \right\}=2{{\cos }^{2}}\left( \frac{\pi }{4}+\alpha  \right)\], which is given in (c).

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