Chứng minh rằng trong mọi tam giác ta đều có :
$\tan \frac{A}{2} + \tan \frac{B}{2} + \tan \frac{C}{2} = \frac{3 + \cos A + \cos B + \cos C}{\sin A + \sin B + \sin C}$.
\[\begin{array}{rcl}
\frac{{3 + \cos A + \cos B + \cos C}}{{\sin A + \sin B + \sin C}} &=& \frac{{\left( {1 + \cos A} \right) + \left( {1 + \cos B} \right) + \left( {1 + \cos C} \right)}}{{2\sin \frac{{A + B}}{2}\cos \frac{{A - B}}{2} + 2\sin \frac{C}{2}\cos \frac{C}{2}}}\\
&=& \frac{{2\left( {{{\cos }^2}\frac{A}{2} + {{\cos }^2}\frac{B}{2} + {{\cos }^2}\frac{C}{2}} \right)}}{{2\cos \frac{C}{2}\left( {\cos \frac{{A - B}}{2} + \cos \frac{{A + B}}{2}} \right)}}\\
&=& \frac{1}{2}.\frac{{{{\cos }^2}\frac{A}{2} + {{\cos }^2}\frac{B}{2} + {{\cos }^2}\frac{C}{2}}}{{\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}}}\\
&=& \frac{1}{2}\left( {\frac{{\sin \frac{{B + C}}{2}}}{{\cos \frac{B}{2}\cos \frac{C}{2}}} + \frac{{\sin \frac{{C + A}}{2}}}{{\cos \frac{C}{2}\cos \frac{A}{2}}} + \frac{{\sin \frac{{A + B}}{2}}}{{\cos \frac{A}{2}\cos \frac{B}{2}}}} \right)\\
&=& \tan \frac{A}{2} + \tan \frac{B}{2} + \tan \frac{C}{2}
\end{array}\]