8sinx=4+cosx
=>(8sinx-4)^2=cos^2x
=>64sin^2x-64sinx+16=1-sin^2x
=>65sin^2x-64sinx+15=0
=>65sin^2x-39sinx-25sinx+15=0
=>13sinx(5sinx-3)-5(5sinx-3)=0
=>(5sinx-3)(13sinx-5)=0
So sinx =3/5 and sin x= 5/13
When sinx =3/5 then cosx =pmsqrt(1-sin^2x)=pm4/5
sinx=3/5 and cosx=4/5 when #x
in " 1st quadrant "
If we put these two values in the given equation we get
LHS=8sinx=8xx3/4=24/5
and
RHS=4+cosx=4+4/5=24/5
Here LHS=RHS
So we can say that sinx =3/5 satisfies the given equation and this value is an acceptable solution when angle x is in 1st quadrant,
If x is in 2nd quadrant the sinx =3/5 but cosx= -4/5, these values do not satisfy the given equation
Again
When sinx =5/13 then cosx =pmsqrt(1-sin^2x)=pm12/13
sinx =5/13 and cosx =12/13 when x in " 1st quadrant"
If we put these two values in the given equation we get
LHS=8sinx=8xx5/13=40/13
and
RHS=4+cosx=4+12/13=64/13
Here LHS!=RHS
But if x is in 2nd quadrant the sinx =5/13 but cosx= -12/13, these values do satisfy the given equation
Hence solution color(red)(tosinx =3/5" when x is in 1st qudrant")
and color(red)(tosinx =5/13" when x is in 2nd quadrant")