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

  • question_answer
    If \[\sin 2\theta +\sin 2\varphi =1/2\] and \[\cos 2\theta +\cos 2\varphi =3/2\], then \[{{\cos }^{2}}(\theta -\varphi )=\] [MP PET 2000; Pb. CET 2000]

    A) \[3/8\]

    B) \[A+B+C=\pi ,\]

    C) \[3/4\]

    D) \[5/4\]

    Correct Answer: B

    Solution :

    Given \[\sin 2\,\theta +\sin 2\varphi =1/2\] ?..(i) and \[\cos 2\,\theta +\cos 2\,\varphi =3/2\] ?..(ii) Square and adding, \[\therefore \,({{\sin }^{2}}2\theta +{{\cos }^{2}}2\theta )+({{\sin }^{2}}2\varphi +{{\cos }^{2}}2\varphi )\]\[+2\,[\sin 2\,\theta \,\sin 2\,\varphi +\cos 2\,\theta \,\cos 2\,\varphi ]=1/4+9/4\] Þ \[\cos 2\theta \cos 2\,\varphi +\sin 2\theta \sin 2\varphi =1/4\] Þ \[\cos (2\theta -2\varphi )=1/4\] Þ \[{{\cos }^{2}}(\theta -\varphi )=5/8\].

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