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Math Help - verify the following identity

  1. #1
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    verify the following identity

    please can someone help me on this problem.

    sinx+sin2x+sin3x+sin4x+sin5x  divided by<br />
cosx+cos2x+cos3x+cos4x+cos5x = tan 3x
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  2. #2
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    Hello,
    Quote Originally Posted by jab1 View Post
    please can someone help me on this problem.

    sinx+sin2x+sin3x+sin4x+sin5x  divided by<br />
cosx+cos2x+cos3x+cos4x+cos5x = tan 3x
    \sin(a)+\sin(b)=2 \sin \left(\frac{a+b}{2}\right)\cos \left(\frac{a-b}{2}\right)

    \cos(a)+\cos(b)=2 \cos \left(\frac{a+b}{2}\right)\cos \left(\frac{a-b}{2}\right)


    \sin(x)+\sin(2x)+\sin(3x)+\sin(4x)+\sin(5x)
    =(\sin(x)+\sin(5x))+(\sin(2x)+\sin(4x))+\sin(3x)
    Using the formula :
    =2 \sin(3x) \cos(x)+2 \sin(3x) \cos(3x)+\sin(3x)
    =\sin(3x) \left(2 \cos(x)+2 \cos(3x)+1\right)


    Similarly, we get :
    \cos(x)+\cos(2x)+\cos(3x)+\cos(4x)+\cos(5x)
    =\cos(3x) \left(2 \cos(x)+2 \cos(3x)+1\right)


    Which simplifies
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  3. #3
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    thank you very much
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