`tan(2x)-cot(x)=0`

express in terms of sin and cos,

`sin(2x)/cos(2x)-cos(x)/sin(x)=0`

`(sin(x)sin(2x)-cos(x)cos(2x))/(cos(2x)sin(x))=0`

`sin(x)sin(2x)-cos(x)cos(2x)=0`

`-(cos(x)cos(2x)-sin(x)sin(2x))=0`

using the identity `cosAcosB-sinAsinB=cos(A+B)`

`rArr-cos(x+2x)=0`

`rArrcos(3x)=0`

General solutions for cos(3x)=0 are,

`3x=pi/2+2pin, x=(3pi)/2+2pin`

`x=(4pin+pi)/6 , x=(4pin+3pi)/6`

Solutions for the range `0<=x<=2pi` are,

`x=pi/6,pi/2,(5pi)/6,(7pi)/6,(3pi)/2,(11pi)/6`

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`tan(2x)-cot(x)=0`

express in terms of sin and cos,

`sin(2x)/cos(2x)-cos(x)/sin(x)=0`

`(sin(x)sin(2x)-cos(x)cos(2x))/(cos(2x)sin(x))=0`

`sin(x)sin(2x)-cos(x)cos(2x)=0`

`-(cos(x)cos(2x)-sin(x)sin(2x))=0`

using the identity `cosAcosB-sinAsinB=cos(A+B)`

`rArr-cos(x+2x)=0`

`rArrcos(3x)=0`

General solutions for cos(3x)=0 are,

`3x=pi/2+2pin, x=(3pi)/2+2pin`

`x=(4pin+pi)/6 , x=(4pin+3pi)/6`

Solutions for the range `0<=x<=2pi` are,

`x=pi/6,pi/2,(5pi)/6,(7pi)/6,(3pi)/2,(11pi)/6`