In (0, π), the number of solutions of the equation tanx+ tan 2x + tan 3x = tan xtan2x.tan3x is

Question

In (0,π), the number of solution of the equation tanx+tan2x+tan3x=tanxtan2xtan3x is

Answer 1

 tan x + tan 2x + tan 3x - tan x tan 2x tan 3x = 0 

=> tan x + tan 2x + tan 3x ( 1 - tan x tan 2x ) = 0 

=> tan x + tan 2x = - tan 3x ( 1 - tan x tan 2x ) 

=> ( tan x + tan 2x )/ ( 1 - tan x tan 2x ) = - tan 3x 

=> tan 3x = - tan 3x 

=> 2 tan 3x = 0 

=> tan 3x = 0 

=> 3x = 2kπ where k = 0, 1,2,3,4,5,.............. 

=> x = 2 k π/3 where k = 0 , 1, 2 ,3 , 4 , 5 , 6..............