# Math Help - atriangle

1. ## 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

2. 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.

3. thank you