Solution:

Given equation: sin(3x)sin(2x) - sin(x) = 4sin(x)cos(x)/2cos(3x)/2

To solve this equation, we will simplify the expression step by step.

1. Simplifying the right side of the equation:
4sin(x)cos(x)/2cos(3x)/2 can be simplified as 2sin(x)cos(x)/cos(3x).

2. Applying the double-angle formula for sine:
sin(2x) = 2sin(x)cos(x).

Substituting this in the equation, we get:
sin(3x) * 2sin(x)cos(x) - sin(x) = 2sin(x)cos(x)/cos(3x).

3. Multiplying through by 2cos(3x):
2sin(3x)sin(x)cos(x) - 2sin(x)cos(3x) = 2sin(x)cos(x).

4. Rearranging the terms:
2sin(3x)sin(x)cos(x) - 2sin(x)cos(x) = 2sin(x)cos(3x).

5. Factoring out 2sin(x):
2sin(x)[sin(3x)cos(x) - 1] = 2sin(x)cos(3x).

6. Dividing both sides by 2sin(x):
sin(3x)cos(x) - 1 = cos(3x).

7. Moving -1 to the right side:
sin(3x)cos(x) = cos(3x) + 1.

8. Applying the double-angle formula for cosine:
cos(2x) = 2cos^2(x) - 1.

Substituting this in the equation, we get:
sin(3x)cos(x) = 2cos^2(3x) - 1 + 1.

9. Simplifying:
sin(3x)cos(x) = 2cos^2(3x).

10. Applying the identity sin(2A) = 2sin(A)cos(A):
sin(3x)cos(x) = sin(6x).

11. Cancelling out sin(x) on both sides:
cos(x) = sin(6x).

12. Taking the square of both sides:
cos^2(x) = sin^2(6x).

13. Using the identity cos^2(x) = 1 - sin^2(x):
1 - sin^2(x) = sin^2(6x).

14. Rearranging the terms:
sin^2(x) + sin^2(6x) = 1.

15. Applying the Pythagorean identity:
sin^2(x) + (1 - cos^2(6x)) = 1.

16. Simplifying:
sin^2(x) + 1 - cos^2(6x) = 1.

17. Rearranging the terms:
sin^2(x) - cos^2(6x) = 0.

18. Applying the identity cos(2A) = cos^2(A) - sin^2(A):
sin^2(x) - cos^2(3x)