Question 195863
Solve for x in the equation {{{2sin(x)tan(x) + tan(x) - 2sin(x) - 1 = 0}}} 

for {{{0<=x<2pi}}} 
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{{{2sin(x)tan(x) + tan(x) - 2sin(x) - 1 = 0}}}

Factor {{{tan(x)}}} out of the first two terms on the left:

{{{tan(x)(2sin(x) + 1) - 2sin(x) - 1 = 0}}}
        
Factor {{{-1}}} out of the last two terms on the left:

{{{tan(x)(2sin(x) + 1)-1(2sin(x) + 1) = 0}}}

Factor {{{(2sin(x)+1)}}} out of the left side:

{{{(2sin(x)+1)(tan(x)-1) = 0}}}

Set each factor = 0

{{{2sin(x)+1=0)}}}        |{{{tan(x)-1 = 0}}}

{{{2sin(x)=-1}}}         |{{{tan(x) = 1}}}

{{{sin(x) = -1/2}}}         |{{{x =pi/4}}}, {{{x = 5pi/4}}}
 
{{{x=7pi/6}}}, {{{x=11pi/6}}}

Solutions, smallest to largest:

{{{pi/4}}}, {{{7pi/6}}}, {{{5pi/4}}}, {{{11pi/6}}} 

Edwin</pre>