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Homework Help: Trigonometry I

  1. Jun 22, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove
    [tex]\frac{sin 2 \alpha}{1+cos 2 \alpha}-\frac{cos \alpha}{1+cos \alpha}=tan \frac{ \alpha}{2}[/tex]




    Use half and double angle formulas, I get;

    The given expression is equal to:
    [tex]\frac{(2 tan \frac{\alpha}{2})(1+tan^{2} \frac{\alpha}{2})}{1-tan^{2}\frac{\alpha}{2}}+2 tan \frac{\alpha}{2}+tan^{2}\frac{\alpha}{2}-1[/tex]

    Please help me simplify this to only [tex]tan \frac{\alpha}{2}[/tex]
     
    Last edited: Jun 22, 2008
  2. jcsd
  3. Jun 22, 2008 #2
    That form looks quite complicated.

    Why don't you try expanding the RHS instead of the LHS using the half angle formula
    [tex]tan \frac{\alpha}{2}= \pm \sqrt_{\frac{1-cos \alpha}{1+cos \alpha} [/tex] [tex]= \frac{sin\alpha}{1+cos\alpha}[/tex]
     
    Last edited: Jun 22, 2008
  4. Jun 22, 2008 #3
    This implies that:
    [tex]2 tan \alpha - \frac{cos \alpha}{1+ cos \alpha}=\frac{sin \alpha}{1+ cos \alpha}[/tex] Isnt it?

    But now the denominators are different. How will I proceed?

    Thanks
     
  5. Jun 22, 2008 #4
    Oh darn. I didn't turn out like I hoped it would. Lol. Let's try your original method.

    Let's see, by the double-angle formula,
    [tex]tan(2\alpha)=\frac{2tan \alpha}{1-tan^2 \alpha} [/tex] (*)

    therefore, your original LHS
    then for the [tex]\frac{(2 tan \frac{\alpha}{2})(1+tan^{2} \frac{\alpha}{2})}{1-tan^{2}\frac{\alpha}{2}}[/tex] term


    By (*) [tex]1-tan^2 \frac{\alpha}{2}= \frac{2tan \frac{\alpha}{2}}{tan \alpha}[/tex], so the [tex]\frac{(2 tan \frac{\alpha}{2})[/tex] should cancel

    Edit:
    This simplifies the LHS into
    [tex]tan \alpha \left( 1+tan^2 \frac{\alpha}{2} \right) + +2 tan \frac{\alpha}{2}+tan^{2}\frac{\alpha}{2}-1[/tex]
     
    Last edited: Jun 22, 2008
  6. Jun 22, 2008 #5

    Defennder

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    Did you transcribe the question correctly? It appears that the "identity" doesn't hold for [tex]\alpha = \frac{\pi}{4}[/tex]
     
  7. Jun 23, 2008 #6

    tiny-tim

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    Hi ritwik06! :smile:
    ooh! :cry:

    Golden rule: whnever you see (1 + cos) or (1 - cos), use the standard trigonometric identities for them! :smile:
     
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