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Math Help - trig proof help

  1. #1
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    trig proof help

    sinx/(1+cosx)
    simplify that so it equals:
    tan(x/2)

    (1-cosx)/sinx
    simplify that so it equals:
    tan(x/2)

    so can someone post the steps please and thanks
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  2. #2
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    Re: trig proof help

    Quote Originally Posted by ahmedb View Post
    sinx/(1+cosx)
    simplify that so it equals:
    tan(x/2)

    (1-cosx)/sinx
    simplify that so it equals:
    tan(x/2)

    so can someone post the steps please and thanks
    start with the basic ratio identity for tangent ...

    \tan\left(\frac{x}{2}\right) = \frac{\sin\left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)}

    you will also need the half-angle identities for sine ...

    \sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1-\cos{x}}{2}}

    and cosine ...

    \cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1+\cos{x}}{2}}
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  3. #3
    MHF Contributor Siron's Avatar
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    Re: trig proof help

    You can also use the t-formulas.
    Let \tan\left(\frac{x}{2}\right)=t then
    \sin(x)=\frac{2t}{1+t^2}
    \cos(x)=\frac{1-t^2}{1+t^2}
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  4. #4
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    Re: trig proof help

    no i am not talking about this.
    i am talking about,
    sin(x)/(1+cos(x))

    when you simplify that you get tan(x/2)
    so how do you get tan(x/2) from the other equation?
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  5. #5
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    Re: trig proof help

    Quote Originally Posted by ahmedb View Post
    no i am not talking about this.
    i am talking about,
    sin(x)/(1+cos(x))

    when you simplify that you get tan(x/2)
    so how do you get tan(x/2) from the other equation?
    Use the double angle identities to express sin and cos in terms of their half-angles since \sin(x) = \sin \left[2\left(\frac{x}{2}\right)\right]

    Let \theta = \dfrac{x}{2} and we arrive at:

    \dfrac{2\sin(\theta)\cos(\theta)}{1+2\cos^2 (\theta) -1} = \dfrac{2\sin \theta \cos \theta}{2\cos^2 \theta}
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  6. #6
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    Re: trig proof help

    i still don't kind of get it
    can you show the steps?
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  7. #7
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    Re: trig proof help

    I still need help
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  8. #8
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    Re: trig proof help

    Quote Originally Posted by ahmedb View Post
    I still need help
    \sin (x) = \sin \left( {2\cdot\tfrac{x}{2}} \right) = 2\sin \left( {\tfrac{x}{2}} \right)\cos \left( {\tfrac{x}{2}} \right).

    In a similar way we get \cos (x) = \cos \left( {2\cdot\tfrac{x}{2}} \right) = 2\cos^2 \left( {\tfrac{x}{2}} \right)-1.

    Use those two in \frac{\sin(x)}{\cos(x)+1} it falls right out.
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  9. #9
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    Re: trig proof help

    so what do you do with the (+1)?
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  10. #10
    Super Member Quacky's Avatar
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    Re: trig proof help

    Have you tried the suggested substitution in post #8? If you have, you'll soon see what happens to the +1.
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