Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\dfrac{a}{{\sin A}} = \dfrac{b}{{\sin B}} = \dfrac{c}{{\sin C}} = 2R\\
\Rightarrow \left\{ \begin{array}{l}
a = 2R.\sin A\\
b = 2R.\sin B\\
c = 2R.\sin C
\end{array} \right.\\
\left( {\tan A + \tan B + \tan C} \right).R\\
= \left( {\dfrac{{\sin A}}{{\cos A}} + \dfrac{{\sin B}}{{\cos B}} + \dfrac{{\sin C}}{{\cos C}}} \right).R\\
= \dfrac{{\sin A.R}}{{\cos A}} + \dfrac{{\sin B.R}}{{\cos B}} + \dfrac{{\sin C.R}}{{\cos C}}\\
= \dfrac{a}{{2\cos A}} + \dfrac{b}{{2\cos B}} + \dfrac{c}{{2\cos C}}\\
= \dfrac{a}{{2.\dfrac{{{b^2} + {c^2} - {a^2}}}{{2bc}}}} + \dfrac{b}{{2.\dfrac{{{c^2} + {a^2} - {b^2}}}{{2ca}}}} + \dfrac{c}{{2.\dfrac{{{a^2} + {b^2} - {c^2}}}{{2ab}}}}\\
= \dfrac{{abc}}{{{b^2} + {c^2} - {a^2}}} + \dfrac{{abc}}{{{c^2} + {a^2} - {b^2}}} + \dfrac{{abc}}{{{a^2} + {b^2} - {c^2}}}
\end{array}\)
