tan x + ( 2 tan x ) / ( 1 - tan² x ) - ( tan x + tan 2 x ) / ( 1 - tan x * tan 2 x ) = 0
[tex] \frac{tanx-tan^{3}x+2tanx }{1-tan ^{2} x} - \frac{tanx-tan ^{3}x+2tanx }{1-tan ^{2}x-2tan ^{2} x }=0 \\ \frac{3tanx-tan ^{3}x }{1-tan ^{2} x} - \frac{3tanx-tan ^{3}x }{1-3tan ^{2}x }=0 [/tex]
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
