in-ABC-prove-that-a-b-c-a-b-c-tan-A-2-tan-B-2-

Question Number 43874 by peter frank last updated on 16/Sep/18

i n Δ A B C, p r o v e t h a t

\frac{a + b - c}{a + b + c} = \tan \frac{A}{2} \tan \frac{B}{2}

Answered by ajfour last updated on 16/Sep/18

\frac{a + b - c}{a + b + c} = \frac{\sin A + \sin B - \sin (A + B)}{\sin A + \sin B + \sin (A + B)}

= \frac{2 \sin (\frac{A + B}{2}) [\cos (\frac{A - B}{2}) - \cos (\frac{A + B}{2})]}{2 \sin (\frac{A + B}{2}) [\cos (\frac{A - B}{2}) + \cos (\frac{A + B}{2})]}

= \frac{2 \sin \frac{A}{2} \sin \frac{B}{2}}{2 \cos \frac{A}{2} \cos \frac{B}{2}} = \tan \frac{A}{2} \tan \frac{B}{2} .

Commented by peter frank last updated on 17/Sep/18

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