`tan(x + pi) + 2sin(x + pi) = 0` Find all solutions of the equation in the interval [0, 2pi).

`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` 

solving`1-2cos(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`

`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` 

solving`1-2cos(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`