摘要或動機

本研究中，我們將提出一些新穎結果，著重討論其在三角中的應用；同時，找出其遞迴關係式，得出三角展開式與其所對應之多項式分解式，進而討論出多種的規律性及所涵蓋的內容及推廣性質，我得到很多高中數學公式無法推導出在【4】和【8】中的漂亮公式及創新的結果，且這些等式都是由我們不太瞭解的無理數所構成的。

主要是討論我們在【7】中所得到的收穫與經驗；複數是三角、幾何、代數互動的橋樑，我是以不同的角度及嶄新的方法來綜合探討在【6】中相關的應用。提出關於正整數平方的倒數和公式更為精簡且基本的證明，將 sin⁻²
x 表示成級數形式的部分分式，進而應用在(a,b) = 1的機率問題上；並研究相關的等式，直接透過三角與代數來研究關於 2p 次方的倒數之求和問題，得出級數
之和的有用遞迴公式，並與最重要的常數扯上關係。



For one thing, we present diverse methods to evaluate finite trigonometric summation
and related sums. Trigonometric summations over the angles equally divided on the
upper half plane are investigated systematically. Several related trigonometric
identities are also exhibited.


What is more, we use methods of calculus, and make several surprising and unexpected
transformations. A useful recursive formula for obtaining the infinite sums of even
order harmonic series, infinite
sums of a few even order harmonic series, which are calculated using the recursive
formulas, are tabulated for easy references. Furthermore, is there any interesting
results and applications？

Finally, the purpose of this paper is to develop a new proof of

and related identities, but their derivations are more complicated. The following
studies are completed under the instruction of the professor.