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Math Help - using the difference formula

  1. #1
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    using the difference formula

    So Im supposed to use the difference formula to figure out cos(2x). This is what I have so far:

    cos(2x)=cos(3x-x)
    cos(3x-x)=(cos3x)(cosx)+(sin3x)(sinx)

    Im just not sure how to finish solving! (my math homework is done online so it's supposed to be as simplified as possible).
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  2. #2
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    Are you sure you can use only the difference formula? Most people derive it with the sum formula using cos(2x) = cos(x+x)
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  3. #3
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    yep we're supposed to use the difference formula.
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  4. #4
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    In that case you will need these identities:

     \sin(3x) = 3\sin(x) - 4\sin^3(x)
     \cos(3x) = 4\cos^3(x) - 3\cos(x)
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  5. #5
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    Hmmm...tried plugging them in and I'm still still stuck . Would you mind showing me what you'd do after they've been plugged in?
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  6. #6
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     \cos(3x)\cos(x) + \sin(3x)\sin(x) = [4\cos^3(x) - 3\cos(x)] \cos(x) + [3\sin(x) - 4\sin^3(x)] \sin(x)

    =  4\cos^4(x) - 3\cos^2(x) + 3\sin^2(x) - 4\sin^4(x)
    =  4[\cos^4(x) - \sin^4(x)]    - 3[  \cos^2(x) - \sin^2(x)]

    Now notice that  \cos^4(x) - \sin^4(x) can be written as a difference of two squares to  (\cos^2(x) + \sin^2(x) ) (\cos^2(x) - \sin^2(x)) .

    Do you see where to go from here?
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  7. #7
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    yep got it!!! thanks sooo much for all the help!
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