Here are some formulae that you will need:
#color(white)("XXX")sin(A+B)=sin(A)cos(B)+sin(B)cos(A)#
#color(white)("XXXXXX")rarr sin(2x)=2sin(x)cos(x)#
#color(white)("XXX")cos(A+B)=cos(A)cos(B)-sin(A)sin(B)#
#color(white)("XXXXXX")rarr cos(2x)=2 cos^2(x)-1#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)(sin(2x)=2sin(x)cos(x)#
#color(green)(sin(3x)=sin(x)cos(2x)+sin(2x)cos(x))#
#color(white)("XXXX")color(green)(=sin(x)[2cos^2(x)-1]+2sin(x)cos(x)[cos(x)])#
#color(white)("XXXX")color(green)(=2sin(x)cos^2(x)-sin(x)+2sin(x)cos^2(x)#
#color(white)("XXXX")color(green)(=sin(x)[4cos^2(x)-1]#
Given
#color(white)("XXXX")color(blue)(sin(2x))+color(green)(sin(3x))=color(red)(sin(x))#
#color(white)("XXXX")color(blue)(2sin(x)cos(x))+color(green)(sin(x)[4cos^2(x)-1])=color(red)(sin(x))#
#rArr#
#color(white)("XXXX")color(purple)(sin(x)) * [4cos^2(x)+2cos(x)-1]=color(purple)(sin(x)) * [1]#
Either
#{:
(sin(x)=0,color(white)("XX")"or"color(white)("XX"),4cos^2(x)+2cos(x)-1=1),
(rarr x in npi,, 2cos^2(x)+cos(x)-1=0),
(,,cos(x)=(-1+-sqrt(1^2-4(2)(-1)))/(2(2))),
(,,cos(x)=-1" or " 1/2),
(,,x in {pi+n * 2pi, -pi/3+n*(2pi)/3})
:}#
