Question

If A+B+C=
`\pi` show that tan A + tan B +tan C = tan A.tan B.tan C

Answer 1

`a + b + c = \pi \\ = > c= \pi - a - b \\ < = > \tan(c) = \tan(\pi - a - b) = -\tan(a + b)`

Explanation:

we have:

`\tan(a) + \tan(b) + \tan(c) \\ = \tan(a) + \tan(b) - \tan(a + b) \\ = \tan( a) + \tan(b) - ( \tan(a) + \tan(b) )/(1 - \tan(a) \tan(b) ) \\ = ( ( \tan(a) + \tan(b) ) \tan(a) \tan(b) )/( \tan(a) \tan(b) - 1 ) (1)`

we also have:

`\tan(a) \tan(b) \tan(c) \\ = - \tan(a) \tan(b) \tan(a + b) \\ = ( -(\tan( a ) + \tan(b) ) \tan(a) \tan(b) )/(1 - \tan(a) \tan(b) ) \\ = (( \tan(a) + \tan(b)) \tan(a) \tan(b) )/( \tan(a) \tan(b) - 1 ) (2)`

from (1)(2) => proven