Question

Solve : tanx+tan2x-tan3x=0

Answer 1

tan x + ( 2 tan x ) / ( 1 - tan² x ) - ( tan x + tan 2 x ) / ( 1 - tan x * tan 2 x ) = 0

`(tanx-tan^(3)x+2tanx )/(1-tan ^(2) x) - (tanx-tan ^(3)x+2tanx )/(1-tan ^(2)x-2tan ^(2) x )=0 \\ (3tanx-tan ^(3)x )/(1-tan ^(2) x) - (3tanx-tan ^(3)x )/(1-3tan ^(2)x )=0`
Substitution a = tan x:
( 3 a - a³ ) / ( 1 - a² ) - ( 3 a - a³ ) / ( 1 - 3 a² ) = 0
( 3 a - a³ ) * ( 1 - 3 a² ) - ( 3 a - a³ ) * ( 1 - a² ) = 0
3 a - 9 a³ - a³ + 3 a^5 - 3 a + 3 a² + a³ - a^5 = 0
2 a^5 - 6 a³ = 0
2 a³ ( a² - 3 ) = 0
a = 0, a = +/-√3
tan x = 0, tan x = +/- √3;
x 1 = k π,
x 2 = π / 3 + k π,
x 3 = - π / 3 + kπ , k ∈ Z