How Can Double-Angle Identities Simplify Trigonometric Equations?

[MaoRaygo]

Homework Statement

I've been having problems with a lot of identity problems lately, but I've found that I'm especially having issues with problems like this one;
Verify the identity,

1/2 Sin (4x) = 2SinxCosx-4Sin³xCosx

Homework Equations

My professor told me you use Sin(A-B)=SinACosB-CosASinB

The Attempt at a Solution

The best I could do was

1/2 Sin4x=2SinxCox-4Sin³xCos

1/2Sin4x=2(SinxCosx-2sin³)

1/2Sin4x=2Sin(x-4x)=-2Sin(3x)

I know that this is a horrible attempt at solving, but some help would be greatly appreciated.

 

[CAF123]

Homework Statement

I've been having problems with a lot of identity problems lately, but I've found that I'm especially having issues with problems like this one;
Verify the identity,

1/2 Sin (4x) = 2SinxCosx-4Sin³xCosx

Homework Equations

My professor told me you use Sin(A-B)=SinACosB-CosASinB

The Attempt at a Solution

The best I could do was

1/2 Sin4x=2SinxCox-4Sin³xCos

1/2Sin4x=2(SinxCosx-2sin³)

1/2Sin4x=2Sin(x-4x)=-2Sin(3x)

I know that this is a horrible attempt at solving, but some help would be greatly appreciated.

Start with the LHS and write \frac{1}{2} \sin(4x)\,\,\text{as}\,\, \frac{1}{2} \sin(2x +2x) and expand using the addition formula as your professor suggested.