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Math Help - Prove this

  1. #1
    Member kjchauhan's Avatar
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    Prove this

    Please Help me to prove this,


    tan^{-1}\left(\frac{tan2\theta + tanh2\phi}{tan2\theta - tanh2\phi}\right)+tan^{-1}\left(\frac{tan\theta - tanh\phi}{tan\theta + tanh\phi}\right)=tan^{-1}\left(cot\theta coth\phi \right)

    Thank You Very Much....
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Prove this

    Are you sure you typed the formula correctly? I don't believe the two sides are equal. I tried values of theta = 0.1 and phi = 0.2 and I get a result of -1.59 for the left side and +1.55 for the right side.
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  3. #3
    Member kjchauhan's Avatar
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    Re: Prove this

    Yes, I typed correctly as given in the book, may be it was printed wrong in book, thanks ..
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  4. #4
    MHF Contributor Amer's Avatar
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    Re: Prove this

    Quote Originally Posted by kjchauhan View Post
    Please Help me to prove this,


    tan^{-1}\left(\frac{tan2\theta + tanh2\phi}{tan2\theta - tanh2\phi}\right)+tan^{-1}\left(\frac{tan\theta - tanh\phi}{tan\theta + tanh\phi}\right)=tan^{-1}\left(cot\theta coth\phi \right)

    Thank You Very Much....
    take the tan for both sides
    left side
    tan \left(tan^{-1}\left(\frac{tan2\theta + tanh2\phi}{tan2\theta - tanh2\phi}\right)+tan^{-1}\left(\frac{tan\theta - tanh\phi}{tan\theta + tanh\phi}\right)\right)

    then use the identity \tan (a+b) = \frac{\tan a + \tan b }{1 - \tan a \tan b }
    resulting

    \dfrac{\dfrac{tan2\theta + tanh2\phi}{tan2\theta - tanh2\phi} + \dfrac{tan\theta - tanh\phi}{tan\theta + tanh\phi}}{1 - \left(\dfrac{tan2\theta + tanh2\phi}{tan2\theta - tanh2\phi}\right)\left(\dfrac{tan\theta - tanh\phi}{tan\theta + tanh\phi}\right)}

    we have to prove that the previous equal  \cot \theta \coth \phi

    You can make the nominator and denominator with common denominator then cancel it
    Last edited by Amer; July 23rd 2012 at 06:53 PM.
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  5. #5
    Member kjchauhan's Avatar
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    Re: Prove this

    Thank you very much for giving such an important hint..
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  6. #6
    MHF Contributor ebaines's Avatar
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    Re: Prove this

    Quote Originally Posted by ebaines View Post
    Are you sure you typed the formula correctly? I don't believe the two sides are equal. I tried values of theta = 0.1 and phi = 0.2 and I get a result of -1.59 for the left side and +1.55 for the right side.
    Update - the formula is indeed correct. My mistake was in relying on my PC to provide the arctan calculation, and forgetting that it can be plus or minus pi. With that correction the formula does indeed work. Sorry for the earlier post.
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