Note that tan (a+b) = (tan a + tan b) / (1  tan a * tan b). Now if we let a = b = x, then
tan 2x = 2 tan x / (1  (tan x)^2)
tan 2x  tan x = 0
(2 tan x)/(1  (tan x)^2)  tan x = 0
(2 tan x)/(1  (tan x)^2) = tan x
2 tan x = tan x (1  (tan x)^2)
2 tan x = tan x  (tan x)^3
tan x + (tan x)^3 = 0
tan x (1 + (tan x)^2) = 0
Since 1 + (tan x)^2 =/= 0 for any x we can divide both sides by it.
tan x = 0
x = k pi for any integer k.
tan 2x = 2 tan x / (1  (tan x)^2)
tan 2x  tan x = 0
(2 tan x)/(1  (tan x)^2)  tan x = 0
(2 tan x)/(1  (tan x)^2) = tan x
2 tan x = tan x (1  (tan x)^2)
2 tan x = tan x  (tan x)^3
tan x + (tan x)^3 = 0
tan x (1 + (tan x)^2) = 0
Since 1 + (tan x)^2 =/= 0 for any x we can divide both sides by it.
tan x = 0
x = k pi for any integer k.

tan(2x)  tan x = 0 >
(1) tan(2x) = 2 tan x/ [1  tan^2(x)], an identity
(2) tan(2x)  tan x = 2 tan x/ [1  tan^2(x)]  tan x = tan x [2 / {1  tan^2(x)}  1] = 0
(3) tan x [2 / {1  tan^2(x)}  1] = tan x [ 2  1 + tan^2(x)  1] / {1  tan^2(x)} = 0
(4) tan x [ 2  1 + tan^2(x)  1] / {1  tan^2(x)} = tan x [ tan^2(x) ] / {1  tan^2(x)} = 0
(5) For tan x = 0 > x = 0 and 180 deg
(6) For [ tan^2(x) ] / {1  tan^2(x)} = 0 >, also x = 0 and 180 deg.
(1) tan(2x) = 2 tan x/ [1  tan^2(x)], an identity
(2) tan(2x)  tan x = 2 tan x/ [1  tan^2(x)]  tan x = tan x [2 / {1  tan^2(x)}  1] = 0
(3) tan x [2 / {1  tan^2(x)}  1] = tan x [ 2  1 + tan^2(x)  1] / {1  tan^2(x)} = 0
(4) tan x [ 2  1 + tan^2(x)  1] / {1  tan^2(x)} = tan x [ tan^2(x) ] / {1  tan^2(x)} = 0
(5) For tan x = 0 > x = 0 and 180 deg
(6) For [ tan^2(x) ] / {1  tan^2(x)} = 0 >, also x = 0 and 180 deg.

2tan x/(1 – tan²x) – tan x 0= 0
2tan x –tan x + tan³x = tan x(1 + tan²x) = 0
tan x = 0, x = kπ where k is any integer
1 + tan²x = 0 tan²x = –1, no solution
2tan x –tan x + tan³x = tan x(1 + tan²x) = 0
tan x = 0, x = kπ where k is any integer
1 + tan²x = 0 tan²x = –1, no solution

tan2x = 2tanx/(1tan^2x)
2tanx = (1tan^2x) X tanx
2tanx = tanx tan^3x
tan^3x tanx = 0
tanx(tan^x 1 ) = 0
tanx = 0
then x= 0
if tanx = 1, then x= 45deg
2tanx = (1tan^2x) X tanx
2tanx = tanx tan^3x
tan^3x tanx = 0
tanx(tan^x 1 ) = 0
tanx = 0
then x= 0
if tanx = 1, then x= 45deg

tan2x= 2tanx/1tan^2x
Using this , I'm sure you can solve further!!!
Using this , I'm sure you can solve further!!!

C[1] is Integers && (x = 2 pi C[1]  x = pi + 2 pi C[1])