the original problem was:

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

so far i've gotten to:
sin(x)(2sin(x)cos(x)-1)=cos^2(x)-sin^2(x)

Where would I go from here?

you cannot do that!

looks like you factored
sin(3x) - sinx = sinx(sin(2x) - 1)
not true!!!!!!

then sin 90 - sin 30 = sin30(sin60 - 1) Is it???, of course not

we have a formula for sin (3x)
= 3(cos^2 x)(sinx) - sin^3 x
then
sin(3x)-sin(x)=cos(2x)
3(cos^2 x)(sinx) - sin^3 x = 1 - 2sin^2 x
replace cos^2 x with 1 - sin^2x, expand and simplify to get
-4sin^3 x + 2sin^2 x + 2sinx - 1 = 0
2sin^2x(-2sinx + 1) -1(-2sinx + 1) = 0
(-2sinx + 1)(2sin^2x - 1) = 0
sinx = 1/2 or sinx = ± 1/√2

x = 30, 150, 45, 135 degrees

To solve the equation sin(3x) - sin(x) = cos(2x), you have simplified it to sin(x)(2sin(x)cos(x) - 1) = cos^2(x) - sin^2(x).

To continue solving the equation, you can further simplify the right side by using the trigonometric identity cos^2(x) - sin^2(x) = cos(2x).

Start by substituting cos(2x) for cos^2(x) - sin^2(x).

sin(x)(2sin(x)cos(x) - 1) = cos(2x)

Now, you can try to simplify the left side of the equation. Notice that there is a common factor of sin(x) that you can factor out.

sin(x)[2sin(x)cos(x) - 1] = cos(2x)

Next, you can try to simplify the expression inside the bracket.

2sin(x)cos(x) - 1

Now, you can use the double-angle identity for sine to rewrite this expression. The double-angle identity for sine is given by sin(2x) = 2sin(x)cos(x).

So, you can rewrite the expression as:

sin(x)(sin(2x) - 1) = cos(2x)

Now, you can further simplify by rearranging the equation:

sin(x)(sin(2x) - 1) - cos(2x) = 0

At this stage, you have simplified the original equation to sin(x)(sin(2x) - 1) - cos(2x) = 0. You can now continue solving the equation using various techniques, such as factoring, graphing, or applying trigonometric identities, depending on the desired approach or the constraints of the problem.