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Proving Reciprocal Identities

  1. Feb 15, 2017 #1
    1. The problem statement, all variables and given/known data
    (secx+1)/(sin2x) = (tanx)/2cosx-2cos2x)

    2. Relevant equations


    3. The attempt at a solution
    Left Side
    ((1+cosx)/cosx)/2sinxcosx

    ((1+cosx)/cosx) x (1/2sinxcosx)
    cancel the a cosx from both to get
    (1/2sinxcosx)
    This is all I could manage with left side so I tried right side
    Right Side
    (sinx/cosx)/2cosx-2cos2x)
    I'm stuck here. I've been trying to find something to change the denominator of the Right Side but I can't think of anything that will work. If someone could let me know where I am going wrong it would be greatly appreciated!
     
  2. jcsd
  3. Feb 15, 2017 #2

    haruspex

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    What happened to the 1+cos?

    Rather than working each side separately, multiply out to get rid of all the denominators.
     
  4. Feb 15, 2017 #3
    I thought I could cancel a cosx, maybe I cannot. I tried to eliminate the denominator like you said. Here's what I got.
    (secx+1)(2cosx-2cos2x) = (Sin2x)(tanx)

    ((1+cosx)/cosx)(2cosx-2cos2x)=(2sinxcosx)(sinx/cosx)
    Expanding
    ((2cosx-2cos2x+2cos2x-2cos3x)/cosx) = 2
    Moving the cosx under left side to the right side and simplifying
    2cosx-2cos3x = 2cosx
    Does this look right? And if so, where do I go from here?
     
  5. Feb 15, 2017 #4

    haruspex

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    Check the right hand side.
     
  6. Feb 15, 2017 #5
    Woops. I think I should have gotten 2sin2xcosx
    2cosx-2cos3x = 2sin2xcosx
    Does this look right?
     
  7. Feb 15, 2017 #6

    haruspex

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    Yes. Keep simplifying.
     
  8. Feb 15, 2017 #7
    2cosx-2cos3x = 2sin2xcosx
    2cosx(1-cos2x) = 2sin2xcosx
    2cosx(sin2x) = 2sin2xcosx
    2cosxsin2x = 2sin2xcosx
    Does this work?
     
  9. Feb 15, 2017 #8

    haruspex

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    Yes.
    Of course, it is not strictly kosher to start with the thing to be proved and deduce a tautology. You need all the steps to be reversible. They are in this case, but it is cleaner to rewrite it in the more persuasive sequence: start with the tautology and deduce the thing to be proved.
     
  10. Feb 15, 2017 #9
    So I should start with 2sin2xcosx and work backwards? Thank you for all the help by the way.
     
  11. Feb 15, 2017 #10

    haruspex

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    Ideally, yes.
     
  12. Feb 15, 2017 #11
    Ok, I'll try it. I'm going to have to practice these quite a bit more I think.
     
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