Đề bài

Cho tam giác \(ABC\), chứng minh rằng: a)    \(\tan A + \tan B + \tan C = \tan A{\rm{ }}{\rm{. }}\tan B{\rm{ }}{\rm{. }}\tan C\) (với điều kiện tam giác \(ABC\) không vuông)

b)    \(\tan \frac{A}{2}{\rm{ }}{\rm{. }}\tan \frac{B}{2} + \tan \frac{B}{2}{\rm{ }}{\rm{. }}\tan \frac{C}{2} + \tan \frac{C}{2}{\rm{ }}{\rm{. }}\tan \frac{A}{2} = 1\)

Lời giải chi tiết

Trong tam giác \(ABC\), ta có \(A + B + C = \pi \).a) Do \(A + B + C = \pi  \Rightarrow A + B = \pi  - C \Rightarrow \tan \left( {A + B} \right) = \tan \left( {\pi  - C} \right)\)Vì \(\tan \left( {A + B} \right) = \frac{{\tan A + \tan B}}{{1 - \tan A\tan B}}\), \(\tan \left( {\pi  - C} \right) = \tan \left( { - C} \right) =  - \tan C\), nên:\(\tan \left( {A + B} \right) = \tan \left( {\pi  - C} \right) \Rightarrow \frac{{\tan A + \tan B}}{{1 - \tan A\tan B}} =  - \tan C\)\( \Rightarrow \tan A + \tan B =  - \left( {1 - \tan A\tan B} \right)\tan C\)\( \Rightarrow \tan A + \tan B =  - \tan C + \tan A\tan B\tan C \Rightarrow \tan A + \tan B + \tan C = \tan A\tan B\tan C\)Bài toán được chứng minh.b) Ta có:\(A + B + C = \pi  \Rightarrow \frac{{A + B + C}}{2} = \frac{\pi }{2} \Rightarrow \frac{{A + B}}{2} = \frac{\pi }{2} - \frac{C}{2} \Rightarrow \tan \left( {\frac{A}{2} + \frac{B}{2}} \right) = \tan \left( {\frac{\pi }{2} - \frac{C}{2}} \right)\)Do \(\tan \left( {\frac{A}{2} + \frac{B}{2}} \right) = \frac{{\tan \frac{A}{2} + \tan \frac{B}{2}}}{{1 - \tan \frac{A}{2}\tan \frac{B}{2}}}\) và \(\tan \left( {\frac{\pi }{2} - \frac{C}{2}} \right) = \cot \frac{C}{2} = \frac{1}{{\tan \frac{C}{2}}}\), nên: \(\tan \left( {\frac{A}{2} + \frac{B}{2}} \right) = \tan \left( {\frac{\pi }{2} - \frac{C}{2}} \right) \Rightarrow \frac{{\tan \frac{A}{2} + \tan \frac{B}{2}}}{{1 - \tan \frac{A}{2}\tan \frac{B}{2}}} = \frac{1}{{\tan \frac{C}{2}}}\)\( \Rightarrow \left( {\tan \frac{A}{2} + \tan \frac{B}{2}} \right)\tan \frac{C}{2} = 1 - \tan \frac{A}{2}\tan \frac{B}{2} \Rightarrow \tan \frac{A}{2}\tan \frac{B}{2} + \tan \frac{B}{2}\tan \frac{C}{2} + \tan \frac{C}{2}\tan \frac{A}{2} = 1\)Bài toán được chứng minh.