It would be nice if there weren't a trigonometric function of 3x in this equation -- if you had only to solve sin(2x)+cos(4x)=1, for example. Unfortunately, this was not the case, and it becomes necessary to reduce the equation to terms of x only. Begin by multiplying through by cos(3x):

cos(3x)sin(2x)+sin(3x)+cos(3x)cos(4x)-2cos(3x)=0

Some quick theorems:

sin(2x)=2sin(x)cos(x)

cos(2x)=1-2sin²(x)

cos(3x)=cos(2x+x)=cos(2x)cos(x)-sin(2x)sin(x)=

cos(x)-2cos(x)sin²(x)-2sin²(x)cos(x)

sin(3x)=sin(2x+x)=sin(2x)cos(x)+cos(2x)sin(x)=

2cos²(x)sin(x)+sin(x)-2sin³(x)

cos(4x)=cos(2(2x))=1-2sin²(2x)=

1-8sin²(x)cos²(x)


x= 15 or 75