Given Equation:
sin 3x - sin 2x - sin x = 4sin x cos x/2 cos 3x/2

Trigonometric Identities:
Before solving the equation, let's recall some important trigonometric identities that will be used in the process:

1. sin(A + B) = sin A cos B + cos A sin B
2. sin(A - B) = sin A cos B - cos A sin B
3. sin(2A) = 2sin A cos A
4. sin(3A) = 3sin A - 4sin^3 A

Solution:

Step 1: Simplify both sides of the equation using the trigonometric identities mentioned above.

sin 3x - sin 2x - sin x = 4sin x cos x/2 cos 3x/2

Using identity 3, we can rewrite sin 2x as 2sin x cos x:

sin 3x - 2sin x cos x - sin x = 4sin x cos x/2 cos 3x/2

Now, let's simplify the right side of the equation:

4sin x cos x/2 cos 3x/2

Using identity 2, we can rewrite cos 3x/2 as cos x/2:

4sin x cos x/2 cos 3x/2 = 4sin x cos x/2 cos x/2

Step 2: Further simplify both sides of the equation.

We can simplify the left side of the equation by expanding sin 3x using identity 4:

3sin x - 4sin^3 x - 2sin x cos x - sin x = 4sin x cos x/2 cos x/2

Combining like terms, we have:

-4sin^3 x + 2sin x - 2sin x cos x = 4sin x cos x/2 cos x/2

Step 3: Rearrange the equation to isolate the trigonometric functions.

Rearranging the equation, we get:

-4sin^3 x - 2sin x cos x + 2sin x - 4sin x cos x/2 cos x/2 = 0

Step 4: Factor out common terms.

Factoring out sin x, we have:

sin x(-4sin^2 x - 2cos x + 2 - 4cos x/2 cos x/2) = 0

Step 5: Solve for sin x.

Setting each factor equal to zero, we have two possible solutions:

sin x = 0

or

-4sin^2 x - 2cos x + 2 - 4cos x/2 cos x/2 = 0

Step 6: Solve for sin x using the quadratic equation.

To solve the quadratic equation, we can use the quadratic formula:

sin x = (-b ± √(b^2 - 4ac)) / (2a)

In the given equation, a = -4, b = -2, and c = 2 - 4cos x/2 cos x/2.

Substituting the values