Precalculus, Chapter 5, 5.3, Section 5.3, Problem 27

3tan^3(x)=tan(x)
3tan^3(x)-tan(x)=0
tan(x)(3tan^2(x)-1)=0 =>
tan(x)=0 , (3tan^2(x)-1)=0
General solutions for tan(x)=0 are,
x=0+pin
Solutions in the range 0<=x<=2pi are,
x=0 , pi ,2pi
solving (3tan^2(x)-1)=0
3tan^2(x)=1
tan^2(x)=1/3
tan(x)=+-1/sqrt(3)
General solutions for tan(x)=-1/sqrt(3) are,
x=(5pi)/6+pin
solutions in the range 0<=x<=2pi are
x=(5pi)/6 , (11pi)/6
General solutions for tan(x)=1/sqrt(3) are
x=pi/6+pin
solutions in the range 0<=x<=2pi are,
x=pi/6 , (7pi)/6
Therefore solutions are,
x=0 , pi/6 , pi , (5pi)/6 , (7pi)/6 , (11pi)/6 , 2pi