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\[
\sum\limits_{j = 1}^{n - 1} {\sin \frac{{j\pi }}{n}} = \cot \frac{\pi }{{2n}}
\]
\[
\sum\limits_{j = 1}^n {\cos \frac{{\left( {2j - 1} \right)\pi }}{n}} = 0
\]
\[
\sum\limits_{j = 1}^n {\cos \frac{{\left( {2j - 1} \right)\pi }}{{2n + 1}}} = \frac{1}{2}
\]
\[
\sum\limits_{j = 1}^{n - 1} {\sin \frac{{2j^2 \pi }}{n}} = \frac{{\sqrt n }}{2}\left( {1 + \cos \frac{{n\pi }}{2} - \sin \frac{{n\pi }}{2}} \right)
\]
\[
\sum\limits_{j = 1}^{n - 1} {\cos \frac{{2j^2 \pi }}{n}} = \frac{{\sqrt n }}{2}\left( {1 + \cos \frac{{n\pi }}{2} + \sin \frac{{n\pi }}{2}} \right) - 1
\]
\[
\sum\limits_{j = 1}^n {\cot ^2 \frac{{j\pi }}{{2n + 1}}} = \frac{1}{3}n\left( {2n - 1} \right)
\]
\[
\sum\limits_{j = 1}^n {\cot ^2 \frac{{\left( {2j - 1} \right)\pi }}{{4n}}} = n\left( {2n - 1} \right)
\]
\[
\sum\limits_{j = 1}^n {\sec ^2 \frac{{\left( {4j - 3} \right)\pi }}{{4n}}} = 2n^2
\]
\[
\sum\limits_{j = 1}^{\left[ {\frac{{n + 1}}{2}} \right] - 1} {\csc ^2 \frac{{j\pi }}{n}} = \left\{ {\begin{array}{*{20}c}
{\frac{1}{6}\left( {n^2 - 1} \right),} & {n为奇数} \\
{\frac{1}{6}\left( {n^2 - 4} \right),} & {n为偶数} \\
\end{array}} \right.
\]
\[
\sum\limits_{j = 1}^{\left[ {\frac{n}{2}} \right]} {\csc ^2 \frac{{\left( {2j - 1} \right)\pi }}{{2n}}} = \left\{ {\begin{array}{*{20}c}
{\frac{1}{2}\left( {n^2 - 1} \right),} & {n为奇数} \\
{\frac{1}{2}n^2 ,} & {n为偶数} \\
\end{array}} \right.
\]
\[
\sum\limits_{j = 1}^{2n} {\csc ^2 \frac{{j\pi }}{{2n + 1}}} = \frac{4}{3}n\left( {n + 1} \right)
\]
\[
\sum\limits_{j = 1}^n {\tan ^4 \frac{{j\pi }}{{2n + 1}}} = \frac{1}{3}n\left( {2n + 1} \right)\left( {4n^2 + 6n - 1} \right)
\]
\[
\sum\limits_{j = 1}^n {\cot ^4 \frac{{j\pi }}{{2n + 1}}} = \frac{1}{{45}}n\left( {2n - 1} \right)\left( {4n^2 + 10n - 9} \right)
\]
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