How do I solve csc^2x-2cscx = 2 - 4sinx for [0,2pi)?

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#csc^2x-2cscx = 2 - 4sinx# for #[0, 2pi)#

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Answer 1 · 2018-05-19T01:12:52

#csc^2x-2cscx = 2 - 4sinx#

#=>(csc^2x-2cscx) - (2 - 4sinx)=0#

#=>csc^2x(1-2/cscx) - 2(1 - 2sinx)=0#

#=>csc^2x(1-2sinx) - 2(1 - 2sinx)=0#

#=>(csc^2x - 2)(1 - 2sinx)=0#

So #csc^2x=2#

#=>sinx=pm1/sqrt2#

When #sinx=1/sqrt2=sin(pi/4)=sin (pi-pi/4)#

#=>x=pi/4 or (3pi)/4#

When #sinx=-1/sqrt2==sin (pi+pi/4)=sin(2pi-pi/4)#

#=>x=(5pi)/4 or (7pi)/4#

Again when

#1-2sinx=0#

#=>sinx=1/2=sin(pi/6)=sin(pi-pi/6)#

#=>x=pi/6or(5pi)/6#