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Trigonometry - Cosec summation

  1. Oct 19, 2013 #1
    1. The problem statement, all variables and given/known data
    If
    $$\csc\frac{\pi}{32}+\csc\frac{\pi}{16}+\csc\frac{\pi}{8}+\csc\frac{\pi}{4}+\csc\frac{\pi}{2}$$
    has the value equal to ##\cot\frac{\pi}{A}## then find A.
    A)61
    B)62
    C)63
    D)64


    2. Relevant equations



    3. The attempt at a solution
    Writing cosec in terms of sin and taking the LCM to make a common denominator doesn't seem to be of any help.

    I can find the value of each term but that would be tedious and of no use.

    I honestly cannot figure out how should I proceed here.

    Any help is appreciated. Thanks!
     
    Last edited: Oct 19, 2013
  2. jcsd
  3. Oct 19, 2013 #2

    Dick

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    Try showing cot(x)-cot(2x)=csc(2x).
     
  4. Oct 19, 2013 #3

    Simon Bridge

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    ##\csc(\frac{\pi}{2})## appears twice.
    All the angles are successive doublings of ##\pi/32##
     
  5. Oct 19, 2013 #4
    Wow! Thanks a lot Dick! :smile:

    How did you come up with that?

    Very sorry for the typo, its ##\pi/8## instead of the second ##\pi/2##.
     
  6. Oct 19, 2013 #5

    Dick

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    I guessed the series must telescope somehow. So somehow cot(2x) must be related to cot(x) with the difference related to a csc. Seems obvious in retrospect, yes?
     
    Last edited: Oct 19, 2013
  7. Oct 19, 2013 #6
    Yes. I liked the way you came up with cot(x)-cot(2x) and solved the problem in few seconds where I was stuck for a week.

    Thank you again! :)
     
  8. Oct 19, 2013 #7

    Dick

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    You're welcome, but it took me more than a "few seconds". Still keeping that strategy in mind might help in the future. If you've got the sum of a bunch of csc's equaling a cot, then if you can express each csc as a difference of two cot's you might be able to sum the series easily. Substitute any other functions you want for 'csc' and 'cot'.
     
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