let's expand this

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

=> sin(2x)cos(x) + sin(x)cos(2x) - sin(x) = cos^2(x) - sin^2(x)

=> [2sin(x)cos(x)]cos(x) + sin(x)[cos^2(x) - sin^2(x)] - sin(x) = cos^2(x) - sin^2(x)

=> 2sin(x)cos^2(x) + sin(x)[cos^2(x) - sin^2(x)] - sin(x) = cos^2(x) - sin^2(x)

we have a lone sin(x), so let's change everything to sin(x)

=> 2sin(x)(1 - sin^2(x)) + sin(x)[1 - sin^2(x) - sin^2(x)] - sin(x) = 1 - sin^2(x) - sin^2(x)

=> 2sin(x) - 2sin^3(x) + sin(x) - 2sin^3(x) - sin(x) = 1 - 2sin^2(x)

=> 2sin(x) - 2sin^3(x) + sin(x) - 2sin^3(x) - sin(x) - 1 + 2sin^2(x) = 0

now this is ugly, let's try and simplify this a bit

=> -4sin^3(x) + 2sin^2(x) + 2sin(x) - 1 = 0

this is a cubic in sin(x), to cut down on typing for the mean time, let's write sin(x) as y, we get:

-4y^3 + 2y^2 + 2y - 1 = 0 ........ah, that looks a little better

=> -2y^2(2y - 1) + (2y - 1) = 0

=> (2y - 1)(1 - 2y^2) = 0

=> y = 1/2 or y = +/- sqrt(1/2)

but, y = sin(x)

=> sin(x) = 1/2 or sin(x) = +/- sqrt(1/2) = +/- 1/sqrt(2) = +/- sqrt(2)/2

=> x = pi/6, 5pi/6 or x = pi/4, 3pi/4, 5pi/4, 7pi/4

so there are 6 values of x for which this happens