雄中段考題

18.
(1)設\(\alpha,\beta,\gamma\)分別為銳角\(\Delta ABC\)之三內角，證明：\(cot \alpha cot\beta+cot \beta cot \gamma+cot \gamma cot \alpha=1\)
(2)\(\displaystyle \frac{cos50^{\circ}}{sin60^{\circ}sin70^{\circ}}+\frac{cos60^{\circ}}{sin50^{\circ}sin70^{\circ}}+\frac{cos70^{\circ}}{sin50^{\circ}sin60^{\circ}}\)之值為何？

請教第18題的（2）如何用（1）的結果得到答案

\( \cos{50^{\circ}} = - \cos{130^{\circ}} = - \cos{(60^{\circ}+70^{\circ})} = - \cos{60^{\circ}} \cos{70^{\circ}} + \sin{60^{\circ}} \sin{70^{\circ}} \)

所求：\(  \displaystyle  \frac{  \cos{50^{\circ}}  }{ \sin{60^{\circ}} \sin{70^{\circ}}  } + \frac{  \cos{60^{\circ}}  }{ \sin{50^{\circ}} \sin{70^{\circ}} } + \frac{  \cos{70^{\circ}}  }{ \sin{50^{\circ}} \sin{60^{\circ}} }   \)

     \( =  \displaystyle  \frac{ - \cos{60^{\circ}} \cos{70^{\circ}} + \sin{60^{\circ}} \sin{70^{\circ}}  }{ \sin{60^{\circ}} \sin{70^{\circ}}  } + \frac{ - \cos{50^{\circ}} \cos{70^{\circ}} + \sin{50^{\circ}} \sin{70^{\circ}} }{ \sin{50^{\circ}} \sin{70^{\circ}} } + \frac{ - \cos{50^{\circ}} \cos{60^{\circ}} + \sin{50^{\circ}} \sin{60^{\circ}} }{ \sin{50^{\circ}} \sin{60^{\circ}} }  \)

     \( =  \displaystyle - \cot{60^{\circ}} \cot{70^{\circ}} + 1 - \cot{50^{\circ}} \cot{70^{\circ}} + 1 - \cot{50^{\circ}} \cot{60^{\circ}} + 1 = 3 - 1 = 2   \)