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Math Help - atriangle

  1. #1
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    atriangle

    a b c is atriangle such that
    tan (45 -(a/2 ) ) +tan (45 - (b/2) ) +tan (45 -( c/2) ) =1

    find
    cos a+cos b +cos c
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  2. #2
    MHF Contributor red_dog's Avatar
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    I think the equality isn't true in a triangle.
    The equality is equivalent to
    \displaystyle \frac{1-\tan\frac{A}{2}}{1+\tan\frac{A}{2}}+\frac{1-\tan\frac{B}{2}}{1+\tan\frac{B}{2}}+\frac{1-\tan\frac{C}{2}}{1+\tan\frac{C}{2}}=1\Leftrightarr  ow
    \displaystyle \Leftrightarrow \tan\frac{A}{2}\tan\frac{B}{2}+\tan\frac{A}{2}\tan  \frac{C}{2}+\tan\frac{B}{2}\tan\frac{C}{2}+2\tan\f  rac{A}{2}\tan\frac{B}{2}\tan\frac{C}{2}=1 (1)
    In a triangle we have the relation \displaystyle \tan\frac{A}{2}\tan\frac{B}{2}+\tan\frac{A}{2}\tan  \frac{C}{2}+\tan\frac{B}{2}\tan\frac{C}{2}=1
    So (1) becomes \displaystyle 2\tan\frac{A}{2}\tan\frac{B}{2}\tan\frac{C}{2}=0, which is not true.
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  3. #3
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    thank you
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