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Simplifying a trigonometric expression

  1. Jul 20, 2013 #1
    1. The problem statement, all variables and given/known data
    If ##\displaystyle \tan\left(\frac{\pi}{4}+\frac{y}{2}\right)=\tan^3\left( \frac{\pi}{4}+\frac {x}{2} \right)##, prove that $$\frac{\sin y}{\sin x}=\frac{3+\sin^2 x}{1+3\sin^2x}$$


    2. Relevant equations



    3. The attempt at a solution
    $$\tan\left(\frac{\pi}{4}+\frac{y}{2}\right)=\frac{1+\tan (y/2)}{1-\tan(y/2)}=\frac{\cos (y/2)+\sin (y/2)}{\cos (y/2)-\sin(y/2)}=\frac{1+\sin y}{\cos y}$$
    Similarly,$$\tan\left(\frac{\pi}{4}+\frac{x}{2}\right)=\frac{1+\sin x}{\cos x}$$
    Plugging them,
    $$\frac{1+\sin y}{\cos y}=\left(\frac{1+\sin x}{\cos x}\right)^3$$
    Stuck here. Need a few hints to proceed further.

    Any help is appreciated. Thanks!
     
  2. jcsd
  3. Jul 20, 2013 #2

    tiny-tim

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    Hi Pranav-Arora! :smile:
    That's not the only way of simplifying that, is it? :wink:

    (and remember, you want all sins and no coss)
     
  4. Jul 20, 2013 #3
    [tex]\tan\left(\frac{\pi}{4}+\frac{y}{2}\right)=\frac{\sin(\pi/4+y/2)}{\sin(\pi/4-y/2)}[/tex]
    Would that help?
     
  5. Jul 20, 2013 #4

    tiny-tim

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    no, i mean find another simplification for $$\frac{\cos (y/2)+\sin (y/2)}{\cos (y/2)-\sin(y/2)}$$ :wink:
     
  6. Jul 20, 2013 #5
    I think I still cannot follow your hint. Would the following work?
    $$\frac{\cos (y/2)+\sin (y/2)}{\cos (y/2)-\sin(y/2)}=\sqrt{\frac{1+\sin y}{1-\sin y}}$$
     
  7. Jul 20, 2013 #6

    tiny-tim

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    yes!!! :smile:

    keep going!!!!​
     
  8. Jul 20, 2013 #7
    Thanks a lot tiny-tim! I have proved it now. :smile:
     
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