What is the term $prod_{x=1}^{19}{cot(frac{xpi}{40})}$? [closed]












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How to solve this problem? I couldn't understand it.






trigonometry







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closed as off-topic by Arnaud D., amWhy, Brahadeesh, José Carlos Santos, Shailesh Nov 28 at 12:55


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Brahadeesh, José Carlos Santos, Shailesh

If this question can be reworded to fit the rules in the help center, please edit the question.













  • Yes, but what is the problem?
    – Jon
    Nov 28 at 12:25










  • Possible duplicate of Product of cotangents of angles in arithmetic progression
    – Arnaud D.
    Nov 28 at 12:37










  • Start by showing that $cot left(frac{pi(20-n)}{40}right) cot left(frac{pi n}{40}right) = 1$.
    – Winther
    Nov 28 at 12:40
















-1














How to solve this problem? I couldn't understand it.






trigonometry







share|cite|improve this question















closed as off-topic by Arnaud D., amWhy, Brahadeesh, José Carlos Santos, Shailesh Nov 28 at 12:55


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Brahadeesh, José Carlos Santos, Shailesh

If this question can be reworded to fit the rules in the help center, please edit the question.













  • Yes, but what is the problem?
    – Jon
    Nov 28 at 12:25










  • Possible duplicate of Product of cotangents of angles in arithmetic progression
    – Arnaud D.
    Nov 28 at 12:37










  • Start by showing that $cot left(frac{pi(20-n)}{40}right) cot left(frac{pi n}{40}right) = 1$.
    – Winther
    Nov 28 at 12:40














-1












-1








-1







How to solve this problem? I couldn't understand it.






trigonometry







share|cite|improve this question















How to solve this problem? I couldn't understand it.






trigonometry




trigonometry






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 28 at 12:35









Arnaud D.

15.6k52343




15.6k52343










asked Nov 28 at 12:24









Jyothi Krishna Gudi

114




114




closed as off-topic by Arnaud D., amWhy, Brahadeesh, José Carlos Santos, Shailesh Nov 28 at 12:55


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Brahadeesh, José Carlos Santos, Shailesh

If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by Arnaud D., amWhy, Brahadeesh, José Carlos Santos, Shailesh Nov 28 at 12:55


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Brahadeesh, José Carlos Santos, Shailesh

If this question can be reworded to fit the rules in the help center, please edit the question.












  • Yes, but what is the problem?
    – Jon
    Nov 28 at 12:25










  • Possible duplicate of Product of cotangents of angles in arithmetic progression
    – Arnaud D.
    Nov 28 at 12:37










  • Start by showing that $cot left(frac{pi(20-n)}{40}right) cot left(frac{pi n}{40}right) = 1$.
    – Winther
    Nov 28 at 12:40


















  • Yes, but what is the problem?
    – Jon
    Nov 28 at 12:25










  • Possible duplicate of Product of cotangents of angles in arithmetic progression
    – Arnaud D.
    Nov 28 at 12:37










  • Start by showing that $cot left(frac{pi(20-n)}{40}right) cot left(frac{pi n}{40}right) = 1$.
    – Winther
    Nov 28 at 12:40
















Yes, but what is the problem?
– Jon
Nov 28 at 12:25




Yes, but what is the problem?
– Jon
Nov 28 at 12:25












Possible duplicate of Product of cotangents of angles in arithmetic progression
– Arnaud D.
Nov 28 at 12:37




Possible duplicate of Product of cotangents of angles in arithmetic progression
– Arnaud D.
Nov 28 at 12:37












Start by showing that $cot left(frac{pi(20-n)}{40}right) cot left(frac{pi n}{40}right) = 1$.
– Winther
Nov 28 at 12:40




Start by showing that $cot left(frac{pi(20-n)}{40}right) cot left(frac{pi n}{40}right) = 1$.
– Winther
Nov 28 at 12:40










1 Answer
1






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3














begin{align}
prod_{n=1}^{19}cotleft(frac{npi}{40}right)
&=cotleft(fracpi 4right)prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=11}^{19}cotleft(frac{npi}
{40}right)\
&=prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=1}^{9}cotleft(frac{(20-n)pi}{40}right)\
&=prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=1}^{9}cotleft(fracpi 2-frac{npi}{40}right)\
&=prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=1}^{9}tanleft(frac{npi}{40}right)\
&=prod_{n=1}^{9}cotleft(frac{npi}{40}right)tanleft(frac{npi}{40}right)\
&=1
end{align}






share|cite|improve this answer




























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    3














    begin{align}
    prod_{n=1}^{19}cotleft(frac{npi}{40}right)
    &=cotleft(fracpi 4right)prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=11}^{19}cotleft(frac{npi}
    {40}right)\
    &=prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=1}^{9}cotleft(frac{(20-n)pi}{40}right)\
    &=prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=1}^{9}cotleft(fracpi 2-frac{npi}{40}right)\
    &=prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=1}^{9}tanleft(frac{npi}{40}right)\
    &=prod_{n=1}^{9}cotleft(frac{npi}{40}right)tanleft(frac{npi}{40}right)\
    &=1
    end{align}






    share|cite|improve this answer


























      3














      begin{align}
      prod_{n=1}^{19}cotleft(frac{npi}{40}right)
      &=cotleft(fracpi 4right)prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=11}^{19}cotleft(frac{npi}
      {40}right)\
      &=prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=1}^{9}cotleft(frac{(20-n)pi}{40}right)\
      &=prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=1}^{9}cotleft(fracpi 2-frac{npi}{40}right)\
      &=prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=1}^{9}tanleft(frac{npi}{40}right)\
      &=prod_{n=1}^{9}cotleft(frac{npi}{40}right)tanleft(frac{npi}{40}right)\
      &=1
      end{align}






      share|cite|improve this answer
























        3












        3








        3






        begin{align}
        prod_{n=1}^{19}cotleft(frac{npi}{40}right)
        &=cotleft(fracpi 4right)prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=11}^{19}cotleft(frac{npi}
        {40}right)\
        &=prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=1}^{9}cotleft(frac{(20-n)pi}{40}right)\
        &=prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=1}^{9}cotleft(fracpi 2-frac{npi}{40}right)\
        &=prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=1}^{9}tanleft(frac{npi}{40}right)\
        &=prod_{n=1}^{9}cotleft(frac{npi}{40}right)tanleft(frac{npi}{40}right)\
        &=1
        end{align}






        share|cite|improve this answer












        begin{align}
        prod_{n=1}^{19}cotleft(frac{npi}{40}right)
        &=cotleft(fracpi 4right)prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=11}^{19}cotleft(frac{npi}
        {40}right)\
        &=prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=1}^{9}cotleft(frac{(20-n)pi}{40}right)\
        &=prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=1}^{9}cotleft(fracpi 2-frac{npi}{40}right)\
        &=prod_{n=1}^{9}cotleft(frac{npi}{40}right)prod_{n=1}^{9}tanleft(frac{npi}{40}right)\
        &=prod_{n=1}^{9}cotleft(frac{npi}{40}right)tanleft(frac{npi}{40}right)\
        &=1
        end{align}







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 28 at 12:37









        Fabio Lucchini

        7,82311326




        7,82311326















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