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The Derivative of a log of a trig function

  1. Apr 9, 2013 #1
    1. The problem statement, all variables and given/known data
    I came across a question like this on a test today, and it says it would be helpful to simplify before differentiation. However, I could not find ways to simplify (1- cos(x))/(1+cos(x)).


    2. Relevant equations

    ln((1- cos (x))/(1+cos(x)))^7)

    3. The attempt at a solution

    I have tried to multiply by the conjugate of denominator and got (1+2cos(x)+cos(x)^2)/sin(x)^2

    But I still do not see how to tackle this problem.
     
  2. jcsd
  3. Apr 9, 2013 #2

    Curious3141

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    Hint: half-angle formula for cos x.

    (EDITED: my mistake, I assumed you were required to integrate rather than differentiate!)

    The half-angle formula is still a very good idea, after you apply this, apply the laws of logs to simplify further before differentiating.
     
    Last edited: Apr 9, 2013
  4. Apr 9, 2013 #3
    Wouldn't it be easier to use the trigonometric identities for cos^2(x) and sin^2(x)?
     
  5. Apr 9, 2013 #4

    Curious3141

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    How would those help?
     
  6. Apr 9, 2013 #5
    Well, given that tan^2(u)=sin^2(u)/cos^2(u) , shouldn't it be possible to rewrite the equation into tan^2(z) , seeing as we have:

    • sin^2(u) = 1/2 - 1/2 cos(2u)
    • cos^2(u) = 1/2 + 1/2 cos(2u)
     
    Last edited: Apr 9, 2013
  7. Apr 9, 2013 #6

    Curious3141

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    What's "z"?

    The half angle formulae are basically equivalent to the double angle formulae, and they allow an immediate simplification.

    Although, frankly, if differentiation is what's required, I don't think prior simplification helps a whole lot.
     
  8. Apr 9, 2013 #7
    z [itex]\in[/itex] ℝ , didn't want it to get mixed up with x

    What does it simplify to using the half-angle formulae? Using the previously stated identities it should be possible to simplify the expression inside the logarithm to tan(x/2)

    EDIT: That's tan^2(x), not tan(x)!
     
    Last edited: Apr 9, 2013
  9. Apr 9, 2013 #8

    Curious3141

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    Should that be ##\displaystyle \tan^2 \frac{x}{2}##? Because that's what I get with the half-angle formula.

    Further simplification with the laws of logs is trivial.
     
  10. Apr 9, 2013 #9
    Yeah, you're right, forgot to square it in my post. I'll edit it.
     
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