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Prove trigonometric identity and determine a counterexample

  1. Apr 11, 2012 #1
    1. The problem statement, all variables and given/known data
    cos(x-y)cosy-sin(x-y)siny=cosx
    a.try to prove that the equation is an identity
    b. determine a counterexample to show that it is not an identity

    2. Relevant equations
    cos(x-y) = cosxcosy+sinxsiny
    sin(x-y) = sinxcosy-cosxsiny


    3. The attempt at a solution
    a.Left side of equatioin: (cosxcosy+sinxsiny)cosy - (sinxcosy-cosxsiny)siny
    = cosxcosycos2y+sinxcosysinycosy - (sinxcosysiny - cosxsin2y)
    I'm not sure where to go from there ...
    b. how would i go about finding a counterexample?
     
  2. jcsd
  3. Apr 11, 2012 #2
    (cosxcosy+sinxsiny)cosy - (sinxcosy-cosxsiny)siny
    = cosxcosycos2y +sinxcosysinycosy - (sinxcosysiny - cosxsin2y)

    That part is wrong. Once you get that part right, consider trying this out: Factor cos(x) from two of the clusters of terms above and a simplification will happen.


    For the counterexample, just find a value of x and a valuye for y so that the equality doesn't hold.
     
    Last edited: Apr 11, 2012
  4. Apr 12, 2012 #3

    Curious3141

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    Homework Helper

    Parts a and b are mutually exclusive. Either the relation given is an identity, or it is not. If it's an identity, you're supposed to prove it as in part a (in which case you don't have to answer part b). If it's not an identity, you can just provide a single counterexample for part b (in this case, you can't answer part a).

    For this question, it is, in fact an identity. So only part a has an answer.

    You know that cos(A+B) = cosAcosB - sinAsinB.

    Now try letting A = x-y and B = y. What happens?
     
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