Find all real numbers in the interval 02pi that satisfy the

Find all real numbers in the interval [0,2pi] that satisfy the equation

tan 2x - tan x = 0

Please show how to solve the answer is (0, pi, pi/6, 5pi/6)

Solution

tan2x - tanx = 0

tan2x = 2tanx / (1-tan²x)

Substituting the same we get,
2tanx - tanx + tan³x = 0
tanx + tan³x = 0
tanx (1 + tan²x) = 0

Substituting 1 + tan²x as Sec²x, we get

tanx. sec²x = 0
either tanx = 0 or sec²x = 0

tanx = 0 ; x = 0, pi or 2pi

Sec²x = 0 is not defined.
so x= 0,pi or 2pi