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Trigonometry Trigonometric Summation

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,755
Please consider the following equation:

$\displaystyle \sum_{k=1}^{n}\cos^4\left(\frac{k\pi}{2n+1} \right)=\frac{6n-5}{16}$

For this particular equation, which I am trying to prove is true, I have found no way to crack it, even if I let $n=2$ and begin to try to combine the terms together, I end up with the annoying terms $\displaystyle \sin \frac {\pi}{10}$ and $\displaystyle \cos \frac {\pi}{10}$ and I am quite certain that this is not the way to go.

I have referred back to Opalg's great posts at this site to search for ideas, but also to no avail...

Any suggestions are welcome to help me to work this problem.

Thanks in advance.
 
Last edited:

Sudharaka

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
Please consider the following equation:

$\displaystyle \sum_{k=1}^{n}\cos^4\left(\frac{k\pi}{2n+1} \right)=\frac{6n-5}{16}$

For this particular equation, which I am trying to prove is true, I have found no way to crack it, even if I let $n=2$ and begin to try to combine the terms together, I end up with the annoying terms $\displaystyle \sin \frac {\pi}{10}$ and $\displaystyle \cos \frac {\pi}{10}$ and I am quite certain that this is not the way to go.

I have referred back to Opalg's great posts at this site to search for ideas, but also to no avail...

Any suggestions are welcome to help me to work this problem.

Thanks in advance.
Hi anemone, :)

Here's a method that I thought of. This may not be the most elegant method however. :)

Use the power reduction formula for the cosine inside the summation.

\[\sum_{k=1}^{n}\cos^4\theta = \sum_{k=1}^{n}\left(\frac{3 + 4 \cos 2\theta + \cos 4\theta}{8}\right)=\frac{3}{8}\sum_{k=1}^{n}1+ \frac{1}{2}\sum_{k=1}^{n}\cos{2\theta}+\frac{1}{8}\sum_{k=1}^{n}\cos{4\theta}\]

where \(\displaystyle\theta=\frac{k\pi}{2n+1}.\)

Then use Lagrange's trigonometric identity for each summation.

Kind Regards,
Sudharaka.
 
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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,755
Hi Sudharaka,

Wow, your suggestion works great!(Smile)

Thank you so much!:D

-anemone
 

Sudharaka

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
Hi Sudharaka,

Wow, your suggestion works great!(Smile)

Thank you so much!:D

-anemone
You are welcome! Nice to hear that it works; I never actually tried it. :)