Given that \( a + b + c = \pi \) and we want to find the value of \( \tan a + \tan b + \tan c \).
Solution:
Since \( a + b + c = \pi \), we have: \( a + b = \pi – c \).
Therefore,
\( \tan(a + b) = \tan(\pi – c) = -\tan c \)
Also, \(\displaystyle \tan(a + b) = \frac{\tan a + \tan b}{1 – \tan a \tan b} \).
Therefore,
\( \begin{aligned} &\frac{\tan a + \tan b}{1 – \tan a \tan b} = -\tan c \\ &\Rightarrow \tan a + \tan b = -\tan c (1 – \tan a \tan b) \\ &\Rightarrow \tan a + \tan b = -\tan c + \tan a \tan b \tan c \\ &\Rightarrow \tan a + \tan b + \tan c = \tan a \tan b \tan c \\ \end{aligned} \)Thus, the final answer is:
\( \boxed{\tan a + \tan b + \tan c = \tan a \tan b \tan c} \)Please let me know in the comments if you find any errors in this solution.