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Math Help - Proving type question

  1. #1
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    Unhappy Proving type question

    (a-b)/(a+b)=[tan{(A-B)}/2]/[tan{(A+B)}/2]
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  2. #2
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    Hello matsci0000
    Quote Originally Posted by matsci0000 View Post
    (a-b)/(a+b)=[tan{(A-B)}/2]/[tan{(A+B)}/2]
    You don't say so, but I presume that a, b, A, B are sides and angles in a triangle. In which case, using the Sine Rule:

    a=\frac{b\sin A}{\sin B}

    \Rightarrow \frac{a-b}{a+b}= \frac{\frac{b\sin A}{\sin B}-b}{\frac{b\sin A}{\sin B}+b}=\frac{\sin A -\sin B}{\sin A + \sin B}

    Now use the identities \sin A +\sin B=2\sin \tfrac12 (A+B)\cos \tfrac12(A-B) and \sin A -\sin B=2\sin\tfrac12(A-B)\cos\tfrac12(A+B):

    \Rightarrow \frac{a-b}{a+b}= \frac{2\sin\tfrac12(A-B)\cos\tfrac12(A+B)}{2\sin\tfrac12(A+B)\cos\tfrac1  2(A-B)}

    = \frac{\tan\tfrac12(A-B)}{\tan\tfrac12(A+B)}

    Grandad
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