Conversion of a trigonometic function

  1. 1. The problem statement, all variables and given/known data
    How to convert
    [itex]\tan(x)\sin(\frac{x}{2})+\cos(\frac{x}{2})[/itex]
    to
    [itex]\frac{\tan(x)}{\sqrt{2(1-\cos(x))}}[/itex]


    2. Relevant equations



    3. The attempt at a solution
    I can convert it to this form: [itex]\frac{\cos(\frac{x}{2})}{\cos(x)}[/itex]
    [itex]\tan(x)\sin(\frac{x}{2})+\cos(\frac{x}{2})[/itex]
    =[itex]\frac{\sin(x)}{\cos(x)}\sin(\frac{x}{2})+ \cos(\frac{x}{2})[/itex]
    =[itex]\frac{1}{\cos(x)}\left(\sin(x)\sin(\frac{x}{2})+ \cos(x)\cos(\frac{x}{2})\right)[/itex]
    using angle sum and difference identities, we get
    [itex]\left(\sin(x)\sin(\frac{x}{2})+ \cos(x)\cos(\frac{x}{2})\right) = \cos(x - \frac{x}{2}) = \cos(\frac{x}{2})[/itex]
    therefore, we have
    [itex]\tan(x)\sin(\frac{x}{2})+\cos(\frac{x}{2}) = \frac{\cos(\frac{x}{2})}{\cos(x)}[/itex]
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. tiny-tim

    tiny-tim 26,054
    Science Advisor
    Homework Helper

    hi frensel :smile:
    hint: multiply by sin(x)/sin(x), and use the half-angle identities :wink:
     
  4. I got it, thx!

    [itex]\frac{\cos(\frac{x}{2})}{\cos{x}}[/itex]
    [itex] = \frac{\sin(x)}{\sin(x)}\frac{\cos(\frac{x}{2})}{ \cos{x}}[/itex]
    [itex]=\tan(x)\frac{\cos(\frac{x}{2})}{\sin(x)}[/itex]

    using double-angle formula, we have
    [itex]\tan(x)\frac{\cos(\frac{x}{2})}{\sin(x)}[/itex]
    [itex]=\tan(x)\frac{\cos(\frac{x}{2})}{2\sin(\frac{x}{2})\cos(\frac{x}{2})}[/itex]
    [itex]=\tan(x)\frac{1}{2\sin(\frac{x}{2})}[/itex]

    finally, using half-angle formula (assuming [itex]\sin(\frac{x}{2})>0[/itex]), then

    [itex]\tan(x)\frac{1}{2\sin(\frac{x}{2})}[/itex]
    [itex]=\tan(x)\frac{1}{2\sqrt{\frac{1-\cos(x)}{2}}}[/itex]
    [itex]=\frac{\tan(x)}{\sqrt{2(1-\cos(x))}}[/itex]

    Well, although I get the correct result, the calculation is so complicated. Is there any easier way to convert the above trigonometric function?
     
  5. tiny-tim

    tiny-tim 26,054
    Science Advisor
    Homework Helper

    you could work backwards (from the answer) …

    (tan / 2sin1/2) - cos1/2 = … ? :smile:
     
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