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JEE Main & Advanced Mathematics Trigonometric Identities Question Bank Fundamental trigonometrical ratios and functions, Trigonometrical ratio of allied angles

  • question_answer
    The value of  \[6({{\sin }^{6}}\theta +{{\cos }^{6}}\theta )-9({{\sin }^{4}}\theta +{{\cos }^{4}}\theta )+4\] is [MP PET 2001]

    A) -3

    B) 0

    C) 1

    D) 3

    Correct Answer: C

    Solution :

    \[6({{\sin }^{6}}\theta +{{\cos }^{6}}\theta )-9({{\sin }^{4}}\theta +{{\cos }^{4}}\theta )+4\] \[=6[{{({{\sin }^{2}}\theta +{{\cos }^{2}}\theta )}^{3}}-3{{\sin }^{2}}\theta {{\cos }^{2}}\theta ({{\sin }^{2}}\theta +{{\cos }^{2}}\theta )]\]\[-9[{{({{\sin }^{2}}\theta +{{\cos }^{2}}\theta )}^{2}}-2{{\sin }^{2}}\theta {{\cos }^{2}}]+4\] \[=6[1-3{{\sin }^{2}}\theta {{\cos }^{2}}\theta ]-9\,[1-2{{\sin }^{2}}\theta {{\cos }^{2}}\theta ]+4\] \[=6-9+4=1\].


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