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Homework Help: Deriving a trigonometric identity

  1. Jun 8, 2007 #1
    For a homework assignment I'm supposed to prove that sin(x)^2+cos(x)^2=1, using only the following identities (along with algebraic operations):

    sin(-x)=-sin(x)
    cos(-x)=cos(x)
    cos(x+y)=cos(x)cos(y)-sin(x)sin(y)
    sin(x+y)=sin(x)cos(y)+cos(x)sin(y)

    I can't figure this out, because as far as I know the identity can only be derived from the Pythagorean Theorem.

    Any help would be much appreciated.
     
  2. jcsd
  3. Jun 8, 2007 #2

    Hurkyl

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    Well, remember that identities such as

    [tex]\cos (\theta + \varphi) = \cos \theta \cos \varphi - \sin \theta \sin \varphi[/tex]

    are true no matter what the arguments are: you can plug anything you want in for [itex]\theta[/itex] and [itex]\varphi[/itex].
     
  4. Jun 8, 2007 #3
    I've tried this a few times, and I got nowhere. Is there a specific direction to go?
     
  5. Jun 8, 2007 #4

    danago

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    [tex]
    \cos (a \pm b) = \cos (a)\cos (b) \mp \sin (a)\sin (b)
    [/tex]

    try using some value for a and b that will make the left hand side equal 1.
     
  6. Jun 8, 2007 #5
    Thanks very much for everyone's help, I now understand this. cos(x + y) can be written as cos(x + -x), or cos(0).
     
    Last edited: Jun 8, 2007
  7. Jun 9, 2007 #6

    Hurkyl

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    Heh, I was thinking more along the lines of looking for an a and b that makes cos^2 and sin^2 appear on the r.h.s. Same thing either way. :smile:
     
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