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Double Angle Formula

  1. Dec 1, 2016 #1
    1. The problem statement, all variables and given/known data
    Simplify cos^2 8x - sin^2x

    2. Relevant equations

    3. The attempt at a solution
    I thought it would be in the format of cos2x
    But I can't seem to figure it out I tried cos (4 * 2x)

    And I tried to change the sin^2x into 1-cos^2x and I could get any farther.

    Not sure how else to simplify.
  2. jcsd
  3. Dec 1, 2016 #2


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    Try writing it as cos(2*4x), or let u = 4x and then do it with cos(2u).

    Sorry, I rushed through reading your problem. What tools do you have other than the double angle formula? You might be able to write this as a difference of squares first, then apply some identities.

    Do you know what the result should look like? How do you know when it is simple enough?

    Last edited: Dec 1, 2016
  4. Dec 1, 2016 #3

    Ray Vickson

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    What, really, is meant by "simplify"?

    The original result is about as simple as it gets. If you try to express everything in terms of ##\cos(x)## and ##\sin(x)## alone, your expression ##\cos^2 (8x) - \sin^2 x## becomes
    $$ 1-\sin^2 x -64 \cos^2 x + 1344 \cos^4 x - 10752 \cos^6 x + 42240 \cos^8 x\\ - 90112 \cos^{10} x
    +106496 \cos^{12} x -65536 \cos^{14} x +16384 \cos^{16} x $$
    Would you say that expression is simpler than the original one?
  5. Dec 1, 2016 #4
    There is no solution unfortunately it was just a problem given:(
  6. Dec 1, 2016 #5
    Yeah first one is definetly simpler.
  7. Dec 2, 2016 #6
    I have done this problem before, In my book they wanted it to be
  8. Dec 4, 2016 #7


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    use reduction identities
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