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Verifying an Identity

  1. Jun 17, 2010 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations

    All trigonometric identities

    3. 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?
     
  2. jcsd
  3. Jun 17, 2010 #2
    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)
     
  4. Jun 18, 2010 #3

    Mark44

    Staff: Mentor

    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
     
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