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Proving trigonometric identities

  1. Aug 24, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove that:

    (1-tanθ)/(1+tanθ)=(cotθ-1)/(cotθ+1)


    2. Relevant equations

    Trig Identities:

    tanθ= sinθ/cosθ
    cotθ= cosθ/sinθ
    1+tanθ=secθ
    1+cotθ=cosecθ

    3. The attempt at a solution

    These sorts of equations are coming up a lot and I am having trouble understanding what I have to do exactly, I have seen people cross multiply which cannot really work considering we haven't got proof that they equal each other and hence can't cross multiply. I have also attempted changing 1+tanθ to secθ and 1+cotθ to cosecθ but I am having no luck. I know there is a certain way to do it, I am just unsure of what this way is.

    If possible could someone give me a guide on how to do it rather than just hint at things? I am coming up to exams in about two weeks and I don't have time to muck around, I need to make sure that I know everything that could be on the test.

    Thanks in advance for your inputs!!
     
  2. jcsd
  3. Aug 24, 2012 #2

    Mark44

    Staff: Mentor

    The two below aren't identities.
    Which is to be expected, because the actual identies are
    1+tan2θ = sec2θ and

    1+cot2θ = csc2θ
    Start on one side (usually the side that seems most complicated, but that's subjective), and use identities to arrive at what you have on the other side.

    For your problem, one approach would be to write all of the tan and cot functions in terms of sin and cos, and go from there.
     
  4. Aug 24, 2012 #3
    Yeah sorry, I knew that but I was jsut rushing to write this down. Ok I will try that. Now say a different situation comes up say it was : (1-sinx)/(1+sinx)=(1-cosx)/(1+cos). Not sure if that is doable but something along those lines like instead of cot and tan it was cos or sin? what would i do then?
     
  5. Aug 25, 2012 #4

    Mark44

    Staff: Mentor

    First off, you can't just make up something and try to show it's an identity. In this case, your equation is not an identity. To see that, note that if x = 0, the left side value is 1, and the right side value is 0.
     
  6. Aug 25, 2012 #5

    Mark44

    Staff: Mentor

    Moving thread to the Precalc section...
     
  7. Aug 25, 2012 #6
    Wouldn't that be much easier if you write tan as 1/cot?
     
  8. Aug 25, 2012 #7
    Or write cot as 1/tan. Yes, I think so.
     
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