Trigonometric Identity Verification | Simplifying sin(4x) and Solving for x

[themadhatter1]

Homework Statement

sin(4x) = 8cos³(x)sin(x)-4sin(x)cos(x)

Homework Equations

All trigonometric identities

The Attempt at a Solution

I can simplify the right side using the double angle identity to:

sin(4x) = 4sin(2x)cos²(x)-2sin(2x)

However, now I'm not sure what to do. Did I take a step in the wrong direction?

 

[themadhatter1]

Never mind, I found the solution myself, here is my process. I was on the right track.

sin(4x) = 4sin(2x)cos2(x)-2sin(2x)

Pythagorean Identity:

sin(4x) = 4sin(2x)(1-sin²x)-2sin(2x)

FOIL:

sin(4x) = 4sin(2x)-4sin(2x)sin²(x)-2sin(2x)

Half Angle Identity:

sin(4x) = 4sin(2x)-4sin(2x)[(1-cos(2x)/2]-2sin(2x)

simplify:

sin(4x) = 4sin(2x)-[4sin(2x)+4sin(2x)cos(2x)]/(2)-2sin(2x)

simplify more:

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

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

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

Double Angle Identity:

sin(4x) = sin(4x)

 

[Mark44]

You can save yourself a lot of typing by working from the left side to the right.

sin(4x) = sin(2(2x)) = 2sin(2x)cos(2x)
= 4sin(x)cos(x)(cos²(x) - sin²(x))
= 4sin(x)cos(x)(2cos²(x) - 1)
= 8sin(x)cos³(x) - 4sin(x)cos(x)
QED