Precalculus, Chapter 5, 5.4, Section 5.4, Problem 73

tan(x+pi)+2sin(x+pi)=0 , 0<=x<=2pi
using tan(x+pi)=tan(x) , sin(x+pi)=-sin(x)
tan(x)-2sin(x)=0
sin(x)/cos(x)-2sin(x)=0
sin(x)-2sin(x)cos(x)=0
sin(x)(1-2cos(x))=0
solving each part separately,
sin(x)=0 , (1-2cos(x))=0
For sin(x)=0,
General solutions are,
x=0+2pin , x=pi+2pin
solutions for the range 0<=x<=2pi are,
x=0,pi,2pi
solving1-2cos(x)=0
cos(x)=1/2
General solutions for cos(x)=1/2 are,
x=pi/3+2pin, (5pi)/3+2pin
Solutions for the range 0<=x<=2pi are,
x=pi/3 , (5pi)/3
combine all the solutions,
x=0,pi,2pi,pi/3,(5pi)/3