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Math Help - Factor and Simplify: I'm missing a step!

  1. #1
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    Factor and Simplify: I'm missing a step!

    Okay, I'm trying to simplify a trigonometric thing. I say "thing" because it's not an equation, and that has me a little confused.

    Beginning: 2 sinx cosx - 4 sin^{3}x cosx

    So, I'm not given what that is equal to, but told to factor and simplify. I looked up the answer, and they gave me the last two steps.

    Second-to-last: (2 sinx cosx) (1 - 2sin^{2}x)

    Last: (sin2x) (cos2x)

    Okay, getting from the second-to-last step to the last is easy, using the double-angle formulae. I just don't know how to get from the beginning to the second-to-last step! I checked it out in MS Excel, and

    2 sinx cosx - 4sin^{3}x cosx does in fact equal (2 sinx cosx) (1 - 2sin^{2}x)

    ... I just don't know how they got there. Is it that there is some common factor in the first two terms of the original thing? Can I divide each side of a subtraction operation in half or something?
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  2. #2
    o_O
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    2sinxcosx - 4sin^{3}xcosx

    You can factor out 2sinxcosx from that entire expression. Multiply it back in and maybe you'll get what they're doing.:
    \underbrace{2\sin x \cos x}_{\sin 2x} \underbrace{(1 - 2\sin^{2}x)}_{\cos2x}

    Hopefully you'll recognize the expressions for sin2x and cos2x.
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  3. #3
    Hop David
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    2 sinx cosx - 4 sin^{3}x cosx =
    2 sinx cosx - 2 sinx cosx (2 sin^{2}x) =
    2 sinx cosx (1 - 2 sin^{2}x)
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  4. #4
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    Smile

    Oh thanks! Hop David has supplied the steps I was missing.

    Let me reiterate to see if I have it right. First, I can pull out (2 sin^{2}x) from 4 sin^{3}x cosx. That leaves me with 2 sinx cosx on both sides of my subtraction sign. I missed that at first.

    Next, the point I REALLY missed, was that A - AB = A(1-B). It would be so easy to go the other way, but my reverse gears are rusty. Thus I can go from
    2 sinx cosx - 2 sinx cosx (2 sin^{2}x)
    to
    2 sinx cosx (1 - 2 sin^{2}x).

    THEN I can use my double-angle identities. Geez, I've done like twenty problems in my book after that one ... they're all cakewalks by comparison.
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