A) \[\tan A\tan B\tan C\]
B) \[\tan \frac{A+B}{2}+\tan \frac{B+C}{2}\]
C) \[\frac{\tan A+\tan B}{\tan C}\]
D) \[\cot A\cot B\operatorname{cotC}\]
Correct Answer: A
Solution :
| As, \[A+B+C=180{}^\circ \] |
| \[\therefore \] \[A+B=180{}^\circ -C\] |
| \[\Rightarrow \] \[\tan \,\,(A+B)=\tan \,\,(180{}^\circ -C)\] |
| \[\Rightarrow \]\[\frac{\tan A+\tan B}{1-\tan A\tan B}=-\tan C\] |
| \[\Rightarrow \]\[\tan A+\tan B=-\tan C+\tan A\tan B\tan C\] |
| \[\Rightarrow \]\[\tan A+\tan B+\tan C=\tan A+\tan B\tan C\] |
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