Results 1 to 11 of 11

Math Help - Calculus

  1. #1
    Newbie
    Joined
    Aug 2013
    From
    Singapore
    Posts
    6

    Calculus

    Hi this appeared on my differentiation homework. for 0<x< π/2

    Prove 1/2 tan (x/2) + 1/ (2^2) tan (x/ (2^2)) + ..... 1/ (2^n) tan (x/ (2^n)) = 1/ (2^n) cot (x/ (2^n)) - cot x.

    Anyone knows how to solve it?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,407
    Thanks
    1294

    Re: Calculus

    Some brackets to avoid ambiguity would be nice... Is it \displaystyle \begin{align*} \frac{1}{2}\tan{\left( \frac{x}{2} \right) } + \frac{1}{2^2}\tan{ \left( \frac{x}{2^2} \right) } + \dots + \frac{1}{2^n}\tan{ \left( \frac{x}{2^n} \right) } = \frac{1}{2^n}\cot{ \left( \frac{x}{2^n} \right) } - \cot{(x)} \end{align*} or \displaystyle \begin{align*} \frac{1}{2\tan{ \left( \frac{x}{2} \right) } } + \frac{1}{2^2 \tan{ \left( \frac{x}{2^2} \right) } } + \dots + \frac{1}{2^n\tan{ \left( \frac{x}{2^n} \right) } } = \frac{1}{2^n\cot{ \left( \frac{x}{2^n} \right) } } - \cot{(x)}  \end{align*}?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2013
    From
    Singapore
    Posts
    6

    Re: Calculus

    Hi it is the first one....
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Aug 2013
    From
    Singapore
    Posts
    6

    Re: Calculus

    Hi it is \displaystyle \begin{align*} \frac{1}{2}\tan{\left( \frac{x}{2} \right) } + \frac{1}{2^2}\tan{ \left( \frac{x}{2^2} \right) } + \dots + \frac{1}{2^n}\tan{ \left( \frac{x}{2^n} \right) } = \frac{1}{2^n}\cot{ \left( \frac{x}{2^n} \right) } - \cot{(x)} \end{align*} ... Thanks!
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Jul 2012
    From
    INDIA
    Posts
    826
    Thanks
    209

    Re: Calculus

    Calculus-24-aug-13.png
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Aug 2013
    From
    Singapore
    Posts
    6

    Re: Calculus

    Thanks man that's brilliant!
    Quote Originally Posted by ibdutt View Post
    Click image for larger version. 

Name:	24 Aug 13.png 
Views:	16 
Size:	44.3 KB 
ID:	29059
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Jun 2013
    From
    Lebanon
    Posts
    69
    Thanks
    31

    Re: Calculus

    Quote Originally Posted by ibdutt View Post
    Click image for larger version. 

Name:	24 Aug 13.png 
Views:	16 
Size:	44.3 KB 
ID:	29059
    Another idea. Prove the following by an easy induction

    \prod _{k=1}^n \text{cos}\left(\frac{x}{2^k}\right)=\frac{1}{2^n}  \sin x \csc \left(\frac{x}{2^n}\right)

    Then take ln of both sides and differentiate
    Last edited by Idea; August 24th 2013 at 06:09 AM. Reason: typo
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Newbie
    Joined
    Aug 2013
    From
    Singapore
    Posts
    6

    Re: Calculus

    I am sorry could you elaborate further? I don't get the eqn... Thanks!

    Quote Originally Posted by Idea View Post
    Another idea. Prove the following by an easy induction

    \prod _{k=1}^n \text{cos}\left(\frac{x}{2^k}\right)=\frac{1}{2^n}  \sin x \csc \left(\frac{x}{2^n}\right)

    Then take ln of both sides and differentiate
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Jun 2013
    From
    Lebanon
    Posts
    69
    Thanks
    31

    Re: Calculus

    The left side is a product

    \cos \left(\frac{x}{2}\right)\cos \left(\frac{x}{2^2}\right)\cos \left(\frac{x}{2^3}\right)\text{...}\text{..}\cos \left(\frac{x}{2^n}\right)=\frac{1}{2^n} \frac{\sin  x}{\sin  \left(x\left/2^n\right.\right)}
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Newbie
    Joined
    Aug 2013
    From
    Singapore
    Posts
    6

    Re: Calculus

    Ummgh how did you arrive at this equation? I am really lost.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Junior Member
    Joined
    Jun 2013
    From
    Lebanon
    Posts
    69
    Thanks
    31

    Re: Calculus

    You have to prove this formula by induction

    for n=1

    \cos \left(\frac{x}{2}\right)=\frac{1}{2}\frac{\sin  x}{\sin  (x/2)}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: December 13th 2011, 09:11 PM
  2. Replies: 2
    Last Post: June 25th 2010, 10:41 PM
  3. Replies: 1
    Last Post: February 11th 2010, 07:09 AM
  4. Replies: 1
    Last Post: June 23rd 2008, 09:17 AM
  5. Replies: 1
    Last Post: June 7th 2008, 11:47 AM

Search Tags


/mathhelpforum @mathhelpforum