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

  • question_answer
    If \[\cos \theta =\frac{1}{2}\left( a+\frac{1}{a} \right),\]then the value of \[\cos 3\theta \]is [MP PET 2001; Pb. CET 2002]

    A) \[\frac{1}{8}\left( {{a}^{3}}+\frac{1}{{{a}^{3}}} \right)\]

    B) \[\frac{3}{2}\left( a+\frac{1}{a} \right)\]

    C) \[\frac{1}{2}\left( {{a}^{3}}+\frac{1}{{{a}^{3}}} \right)\]

    D) \[\frac{1}{3}\left( {{a}^{3}}+\frac{1}{{{a}^{3}}} \right)\]

    Correct Answer: C

    Solution :

    \[\because \ \cos 3\theta =4{{\cos }^{3}}\theta -3\cos \theta \] \[\therefore \cos 3\theta =4\frac{1}{{{2}^{3}}}{{\left( a+\frac{1}{a} \right)}^{3}}-3\frac{1}{2}\left( a+\frac{1}{a} \right)\] \[\Rightarrow \cos 3\,\theta =\frac{1}{2}\left( a+\frac{1}{a} \right)\,\left[ {{\left( a+\frac{1}{a} \right)}^{2}}-3 \right]\] Þ \[\cos 3\theta =\frac{1}{2}\left( {{a}^{3}}+\frac{1}{{{a}^{3}}} \right)\].

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