To solve this equation, we can first factor out sin(x) from each term:

sin(x) - sin(2x) + sin(3x) - sin(4x) = 0
sin(x)(1 - 2cos(x) + 3 - 4cos(x)) = 0
sin(x)(4 - 6cos(x)) = 0

Now we have two possible solutions:

sin(x) = 0
This occurs when x = nπ, where n is an integer.

4 - 6cos(x) = 0
cos(x) = 2/3
x = arccos(2/3) + 2nπ or x = -arccos(2/3) + 2nπ, where n is an integer.

Therefore, the solutions to the equation sin(x) - sin(2x) + sin(3x) - sin(4x) = 0 are x = nπ and x = arccos(2/3) + 2nπ or x = -arccos(2/3) + 2nπ.