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JEE Main & Advanced Mathematics Trigonometric Identities Question Bank Self Evaluation Test - Trigonometric Function

  • question_answer
    If \[\sin (\pi \cos x)=cos(\pi sinx),\] then what is one of the values of\[sin\text{ }2x\]?

    A) \[-\frac{1}{4}\]

    B) \[-\frac{1}{2}\]

    C) \[-\frac{3}{4}\]             

    D) \[-1\]

    Correct Answer: C

    Solution :

    Given that: \[\sin (\pi \cos x)=\cos (\pi \sin x)\] So,  \[\cos \left( \frac{\pi }{2}-\pi \cos x \right)=\cos (\pi \sin x)\] \[\Rightarrow \,\,\frac{\pi }{2}-\pi \cos x=\pi \sin x\] \[\Rightarrow \,\,\sin x+\cos x=\frac{1}{2}\] Squaring both sides, we get \[{{\sin }^{2}}x+{{\cos }^{2}}x+2\sin x\cos x=\frac{1}{4}\] \[\Rightarrow \,\,\sin 2x=\frac{1}{4}-1=-\frac{3}{4}\]


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