What is the term $prod_{x=1}^{19}{cot(frac{xpi}{40})}$? [closed]
How to solve this problem? I couldn't understand it.
trigonometry
edited Nov 28 at 12:35
Arnaud D.
15.6k52343
asked Nov 28 at 12:24
Jyothi Krishna Gudi
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:
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add a comment |
How to solve this problem? I couldn't understand it.
trigonometry
edited Nov 28 at 12:35
Arnaud D.
15.6k52343
asked Nov 28 at 12:24
Jyothi Krishna Gudi
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.
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
add a comment |
How to solve this problem? I couldn't understand it.
trigonometry
edited Nov 28 at 12:35
Arnaud D.
15.6k52343
asked Nov 28 at 12:24
Jyothi Krishna Gudi
114
How to solve this problem? I couldn't understand it.
trigonometry
trigonometry
edited Nov 28 at 12:35
Arnaud D.
15.6k52343
asked Nov 28 at 12:24
Jyothi Krishna Gudi
114
edited Nov 28 at 12:35
Arnaud D.
15.6k52343
asked Nov 28 at 12:24
Jyothi Krishna Gudi
114
edited Nov 28 at 12:35
Arnaud D.
15.6k52343
edited Nov 28 at 12:35
Arnaud D.
15.6k52343
edited Nov 28 at 12:35
Arnaud D.
15.6k52343
15.6k52343
asked Nov 28 at 12:24
Jyothi Krishna Gudi
114
asked Nov 28 at 12:24
Jyothi Krishna Gudi
114
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
add a comment |
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
add a comment |
1 Answer
1
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oldest
votes
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}
answered Nov 28 at 12:37
Fabio Lucchini
7,82311326
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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}
answered Nov 28 at 12:37
Fabio Lucchini
7,82311326
add a comment |
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}
answered Nov 28 at 12:37
Fabio Lucchini
7,82311326
add a comment |
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}
answered Nov 28 at 12:37
Fabio Lucchini
7,82311326
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}
answered Nov 28 at 12:37
Fabio Lucchini
7,82311326
answered Nov 28 at 12:37
Fabio Lucchini
7,82311326
answered Nov 28 at 12:37
Fabio Lucchini
7,82311326
answered Nov 28 at 12:37
Fabio Lucchini
7,82311326
7,82311326
add a comment |
add a comment |
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