# 雄中段考題

## 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？

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2019-3-5 17:00

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## 回復 1# Exponential 的帖子

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

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

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

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