Help please!

sin4x=3cos^3xsinx-4sin^3xcosx

I'm not sure if this is a half angle identity because the cubes throw me off

simplify the left side into x's

sin 4x
= 2sin 2x cos 2x
= 4sinx cosx(cos^2x - sin^2x)
= 4cos^3 x sinx - 4sin^2 x cosx

= almost the right side

I suspect a typo

To solve the equation sin(4x) = 3cos^3(x)sin(x) - 4sin^3(x)cos(x), you can follow these steps:

Step 1: Use the trigonometric identity sin(2x) = 2sin(x)cos(x) to rewrite sin(4x). This identity states that sin(2x) is equal to twice the product of sin(x) and cos(x).

sin(4x) = 2sin(2x)cos(2x)

Step 2: Apply the double angle formulas for sin(2x) and cos(2x) to express sin(4x) in terms of sin(x) and cos(x).

sin(4x) = 2(2sin(x)cos(x))(cos^2(x) - sin^2(x))

Simplifying further,
sin(4x) = 4sin(x)cos(x)cos^2(x) - 4sin(x)cos(x)sin^2(x)

Step 3: Rearrange the equation to isolate the terms involving sin(x) and cos(x).

sin(4x) = 4cos(x)sin(x)(cos^2(x) - sin^2(x))

Step 4: Use the Pythagorean identity cos^2(x) - sin^2(x) = cos(2x) to simplify the equation further.

sin(4x) = 4cos(x)sin(x)cos(2x)

Step 5: Substitute the expression for sin(4x) from Step 2 into the equation.

2sin(2x)cos(2x) = 4cos(x)sin(x)cos(2x)

Step 6: Divide both sides of the equation by 2cos(2x) to solve for sin(2x).

sin(2x) = 2cos(x)sin(x)

Step 7: Divide both sides of the equation by sin(x) to solve for sin(2x) further.

sin(2x)/sin(x) = 2cos(x)

Step 8: Apply the identity sin(2x) = 2sin(x)cos(x) to rewrite sin(2x)/sin(x).

2cos(x)cos(x)/sin(x) = 2cos(x)

Step 9: Use the identity cos^2(x)/sin(x) = sin(x) to simplify the equation.

2sin(x) = 2cos(x)

Step 10: Divide both sides of the equation by 2 to isolate cos(x) and sin(x).

sin(x) = cos(x)

Now, we have sin(x) = cos(x). To find the x values where this equation holds true, we can use the unit circle.

On the unit circle, the points (sqrt(2)/2, sqrt(2)/2) and (-sqrt(2)/2, -sqrt(2)/2) are such that sin(x) = cos(x). Therefore, we can write the solutions as:

x = π/4 + 2πk, 3π/4 + 2πk where k is an integer.

These are the solutions for the given equation sin(4x) = 3cos^3(x)sin(x) - 4sin^3(x)cos(x).