Question

Tan3x +tan2x+tanx = tanx . tan2x .tan3x please note the LHS operator is plus sign. i.e positive. You have repliced earlier with negative sign point of view. However i got a question from my class teacher in last unit test

Answer 1

To solve for x:     Tan x + tan 2x + tan 3x = tan x * tan 2x * tan 3x 

tan3x = tan(2x+x) = (tan 2x + tan x)/(1- tan 2x tan x)
so  tan x + tan 2x =  tan 3x (1 - tan 2x  tan x)

LHS = tan 3x [ 1 + 1 - tan 2x tan x] = RHS = tan x tan 2x tan 3x
so  simplify to get
               either tan x = 0 ie.,  x = n *π  or,
   or,       tan x tan 2x = 1
  ie.,       tanx * 2 tan x / (1 - tan^2 x) = 1
so           tan^2 x = 1/3             so  x = n * π/3      where n = any integer.

Answer  =   ..., -120, -60, 0, 60, 120, 180, 240, 300, 360,....