tan(4x) = tan(2*(2x)) = 2*tan(2x)/[1 - tan^2(2x)]

So

tan(2x) * tan(4x) = 1

becomes

2*tan^2(2x)/[1 - tan^2(2x)] = 1

2*tan^2(2x) = 1 - tan^2(2x)

3*tan^2(2x) = 1

tan^2(2x) = 1/3

tan(2x) = 1/sqrt{3} <-- Only keep the + solution since 0 <= x < 2(pi)

2x = (pi)/6 (rad)

x = (pi)/12 (rad)

-Dan