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

  • question_answer
    \[{{\cos }^{4}}\frac{\pi }{8}+{{\cos }^{4}}\frac{3\pi }{8}+{{\cos }^{4}}\frac{5\pi }{8}+{{\cos }^{4}}\frac{7\pi }{8}=\]

    A) \[\frac{1}{2}\]

    B) \[\frac{1}{4}\]

    C) \[\frac{3}{2}\]

    D) \[\frac{3}{4}\]

    Correct Answer: C

    Solution :

    \[{{\cos }^{4}}\frac{\pi }{8}+{{\cos }^{4}}\frac{3\pi }{8}+{{\cos }^{4}}\frac{5\pi }{8}+{{\cos }^{4}}\frac{7\pi }{8}\] \[={{\cos }^{4}}\frac{\pi }{8}+{{\cos }^{4}}\frac{3\pi }{8}+{{\cos }^{4}}\frac{3\pi }{8}+{{\cos }^{4}}\frac{\pi }{8}\] \[=2\left( {{\cos }^{4}}\frac{\pi }{8}+{{\cos }^{4}}\frac{3\pi }{8} \right)\] \[=2\left[ {{\left( {{\cos }^{2}}\frac{\pi }{8}+{{\cos }^{2}}\frac{3\pi }{8} \right)}^{2}}-2{{\cos }^{2}}\frac{\pi }{8}{{\cos }^{2}}\frac{3\pi }{8} \right]\] \[=2\left[ 1-\frac{1}{2}\left( 2{{\cos }^{2}}\frac{\pi }{8} \right)\,\left( 2{{\cos }^{2}}\frac{3\pi }{8} \right) \right]\] \[=2-\left( 1+\cos \frac{\pi }{4} \right)\,\left( 1+\cos \frac{3\pi }{4} \right)\] \[=2-\left( 1+\cos \frac{\pi }{4} \right)\,\left( 1-\cos \frac{\pi }{4} \right)\] \[=2-\left( 1-{{\cos }^{2}}\frac{\pi }{4} \right)=2-\left( 1-\frac{1}{2} \right)=2-\frac{1}{2}=\frac{3}{2}\].

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