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Find the value of $n$ in this equation :$$n-1 + \frac{(n-1)}{4}\sum_{i=3}^n \csc\left(\frac{(i-2)\pi}{n-1}\right) \lt \frac{n}{4}\sum_{i=2}^n \csc\left(\frac{(i-1)\pi}{n}\right)$$

My try : I made $f(n) =n-1 + \frac{(n-1)}{4}\sum_{i=3}^n \csc\left(\frac{(i-2)\pi}{n-1}\right) - \frac{n}{4}\sum_{i=2}^n \csc\left(\frac{(i-1)\pi}{n}\right)$ and tried to solve $f(n)\lt 0$ but didn't get any result.

Note: I used Mathematica for solving it and $n=12$ is the first number that satisfies. Also I plotted it but still there is no way to solve it by hand and get the $12$.enter image description here

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    The $n$th term in the sum on the left side is $\csc((n-1)\pi/(n-1))$, which simplifies to $\csc(\pi)$, which is undefined. So the only way I can make sense of this is if $n<2$, in which case the right side is zero and the left side is $n-1$, in which case any $n < 1$ works. Or maybe it's not the correct formula? – 2017-01-18
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    @DavidK Yes , It was my mistake . Sorry – 2017-01-19
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    I have the strong feeling that the present question is related with your previous question (http://math.stackexchange.com/questions/2103259/finding-a-closed-form-expression-for-sum-i-1n-1-csc-fraci-pin) through a common origin. It would be probably easier for us to work on the original problem directly. – 2017-01-19
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    Was it the unbounded-ness of the $L^1$ norm of the Dirichlet kernel (https://en.wikipedia.org/wiki/Dirichlet_kernel), by chance? – 2017-01-19
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    @JackD'Aurizio Yes , it's related but I want to solve this problem and it's coming from a question physics . – 2017-01-19
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    @S.H.W: all right, which one? – 2017-01-19
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    @JackD'Aurizio Is it true now ? – 2017-01-19
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    This problem that presents here . This is original question. – 2017-01-19
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    @S.H.W: in such a case, the problem makes no sense at all. I would expect that such inequality holds for every $n$ in some range or for no $n$s at all, but I cannot wrap my head around the question "what is **the** value of $n$ such that $f(n)< g(n)$ ? ". It is an inequality, not an equation. – 2017-01-19
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    Yes you're right and I want an inequality that show value of $n$. For example $n\lt5$ – 2017-01-19
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    I'm really sorry that I couldn't convey my purpose properly . – 2017-01-19
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    @JackD'Aurizio Is my question right now ? – 2017-01-19

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