Question Number 54472 by Tawa1 last updated on 04/Feb/19
$$\mathrm{If}\:\mathrm{A},\:\mathrm{B},\:\mathrm{C}\:\:\mathrm{are}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle},\:\:\mathrm{show}\:\mathrm{that} \\ $$$$\:\:\:\mathrm{tanA}\:+\:\mathrm{tanB}\:+\:\mathrm{tanC}\:\:=\:\:\mathrm{tanA}\:\mathrm{tanB}\:\mathrm{tanC} \\ $$
Answered by math1967 last updated on 04/Feb/19
$${A}+{B}+{C}=\pi \\ $$$$\Rightarrow{A}+{B}=\pi−{C} \\ $$$$\Rightarrow{tan}\left({A}+{B}\right)={tan}\left(\pi−{C}\right) \\ $$$$\Rightarrow\frac{{tanA}+\mathrm{tan}\:{B}}{\mathrm{1}−{tanA}\mathrm{tan}\:{B}}=−{tanC} \\ $$$$\therefore\mathrm{tan}\:{A}+\mathrm{tan}\:{B}+\mathrm{tan}\:{C}=\mathrm{tan}\:{A}\mathrm{tan}\:{B}\mathrm{tan}\:{C} \\ $$
Commented by Tawa1 last updated on 04/Feb/19
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$