Explanation:

To solve the given equation Sin 3x * Sin 2x - Sin x = 4Sin x * Cos x/2 * Cos 3x/2, we need to simplify both sides of the equation and use trigonometric identities to rearrange the terms.

Simplifying the left side:
Using the trigonometric identity Sin(A-B) = Sin A * Cos B - Cos A * Sin B, we can simplify the left side of the equation as follows:

Sin 3x * Sin 2x - Sin x
= (Sin 3x * Cos 2x) - (Cos 3x * Sin 2x) - Sin x
= Sin(3x - 2x) - Sin(2x - 3x) - Sin x
= Sin x - Sin(-x) - Sin x
= 2Sin x

Therefore, the left side of the equation simplifies to 2Sin x.

Simplifying the right side:
Using the trigonometric identity Cos 2A = 2Cos^2 A - 1, we can simplify the right side of the equation as follows:

4Sin x * Cos x/2 * Cos 3x/2
= 4Sin x * 2Cos^2(x/2) - 1
= 8Sin x * Cos^2(x/2) - 4Sin x

Now, let's substitute the simplified expressions of both sides back into the original equation:

2Sin x = 8Sin x * Cos^2(x/2) - 4Sin x

Applying trigonometric identity:
Using the trigonometric identity 1 - Cos 2A = 2Sin^2 A, we can simplify the equation further:

2Sin x = 8Sin x * (1 - Sin^2(x/2)) - 4Sin x
= 8Sin x - 8Sin^3(x/2) - 4Sin x
= 4Sin x - 8Sin^3(x/2)

Now, let's rearrange the terms:

4Sin x - 8Sin^3(x/2) - 2Sin x = 0
2Sin x (2 - 4Sin^2(x/2)) = 0

Solving for x:
To find the values of x that satisfy the equation, we set each factor equal to zero:

2Sin x = 0
Sin x = 0
x = 0, π, 2π, ...

2 - 4Sin^2(x/2) = 0
4Sin^2(x/2) = 2
Sin^2(x/2) = 1/2
Sin(x/2) = ±√(1/2)
x/2 = π/4, 3π/4, 5π/4, 7π/4, ...

Solving for x, we have:
x = π/2, 3π/2, 5π/2, 7π/2, ...

Therefore, the solutions to the given equation are x = 0, π, 2π, ... and x = π/2, 3π/2, 5π/2, 7

You can simply use the formula