計算\(\displaystyle \frac{1}{sin15^{\circ}sin30^{\circ}}+\frac{1}{sin30^{\circ}sin45^{\circ}}+\frac{1}{sin45^{\circ}sin60^{\circ}}+\ldots+\frac{1}{sin150^{\circ}sin165^{\circ}}\)=？
[解答]
\(\displaystyle =\frac{1}{sin15^{\circ}}\left[\frac{sin(30^{\circ}-15^{\circ})}{sin15^{\circ}sin30^{\circ}}+
\frac{sin(45^{\circ}-30^{\circ})}{sin30^{\circ}sin45^{\circ}}+
\frac{sin(60^{\circ}-45^{\circ})}{sin45^{\circ}sin60^{\circ}}+\ldots
\frac{sin(165^{\circ}-150^{\circ})}{sin150^{\circ}sin165^{\circ}} \right]\)
利用\(\displaystyle \frac{sin(30^{\circ}-15^{\circ})}{sin15^{\circ}sin30^{\circ}}=\frac{sin30^{\circ}cos15^{\circ}-sin15^{\circ}cos30^{\circ}}{sin15^{\circ}sin30^{\circ}}=cot15^{\circ}-cot30^{\circ}\)
原式可改寫成
\(\displaystyle =\frac{1}{sin15^{\circ}}\left[ (cot15^{\circ}-cot30^{\circ})+(cot30^{\circ}-cot45^{\circ})+\ldots+(cot150^{\circ}-cot165^{\circ}) \right]\)
接著後面就交給你了
