三角学示例

\frac{\sin (x)}{1 - \cos (x)} + \frac{\sin (x)}{1 + \cos (x)} = 2 \csc (x)

$\frac{\sin (x)}{1 - \cos (x)} + \frac{\sin (x)}{1 + \cos (x)} = 2 \csc (x)$

解题步骤 1

从左边开始。

\frac{\sin (x)}{1 - \cos (x)} + \frac{\sin (x)}{1 + \cos (x)}

$\frac{\sin (x)}{1 - \cos (x)} + \frac{\sin (x)}{1 + \cos (x)}$

解题步骤 2

加上分数。

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解题步骤 2.1

要将

\frac{\sin (x)}{1 - \cos (x)}

$\frac{\sin (x)}{1 - \cos (x)}$ 写成带有公分母的分数，请乘以

\frac{1 + \cos (x)}{1 + \cos (x)}

$\frac{1 + \cos (x)}{1 + \cos (x)}$ 。

\frac{\sin (x)}{1 - \cos (x)} \cdot \frac{1 + \cos (x)}{1 + \cos (x)} + \frac{\sin (x)}{1 + \cos (x)}

$\frac{\sin (x)}{1 - \cos (x)} \cdot \frac{1 + \cos (x)}{1 + \cos (x)} + \frac{\sin (x)}{1 + \cos (x)}$

解题步骤 2.2

要将

\frac{\sin (x)}{1 + \cos (x)}

$\frac{\sin (x)}{1 + \cos (x)}$ 写成带有公分母的分数，请乘以

\frac{1 - \cos (x)}{1 - \cos (x)}

$\frac{1 - \cos (x)}{1 - \cos (x)}$ 。

\frac{\sin (x)}{1 - \cos (x)} \cdot \frac{1 + \cos (x)}{1 + \cos (x)} + \frac{\sin (x)}{1 + \cos (x)} \cdot \frac{1 - \cos (x)}{1 - \cos (x)}

$\frac{\sin (x)}{1 - \cos (x)} \cdot \frac{1 + \cos (x)}{1 + \cos (x)} + \frac{\sin (x)}{1 + \cos (x)} \cdot \frac{1 - \cos (x)}{1 - \cos (x)}$

解题步骤 2.3

通过与

1

$1$ 的合适因数相乘，将每一个表达式写成具有公分母

(1 - \cos (x)) (1 + \cos (x))

$(1 - \cos (x)) (1 + \cos (x))$ 的形式。

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解题步骤 2.3.1

将

\frac{\sin (x)}{1 - \cos (x)}

$\frac{\sin (x)}{1 - \cos (x)}$ 乘以

\frac{1 + \cos (x)}{1 + \cos (x)}

$\frac{1 + \cos (x)}{1 + \cos (x)}$ 。

\frac{\sin (x) (1 + \cos (x))}{(1 - \cos (x)) (1 + \cos (x))} + \frac{\sin (x)}{1 + \cos (x)} \cdot \frac{1 - \cos (x)}{1 - \cos (x)}

$\frac{\sin (x) (1 + \cos (x))}{(1 - \cos (x)) (1 + \cos (x))} + \frac{\sin (x)}{1 + \cos (x)} \cdot \frac{1 - \cos (x)}{1 - \cos (x)}$

解题步骤 2.3.2

将

\frac{\sin (x)}{1 + \cos (x)}

$\frac{\sin (x)}{1 + \cos (x)}$ 乘以

\frac{1 - \cos (x)}{1 - \cos (x)}

$\frac{1 - \cos (x)}{1 - \cos (x)}$ 。

\frac{\sin (x) (1 + \cos (x))}{(1 - \cos (x)) (1 + \cos (x))} + \frac{\sin (x) (1 - \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}

$\frac{\sin (x) (1 + \cos (x))}{(1 - \cos (x)) (1 + \cos (x))} + \frac{\sin (x) (1 - \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}$

解题步骤 2.3.3

重新排序

(1 - \cos (x)) (1 + \cos (x))

$(1 - \cos (x)) (1 + \cos (x))$ 的因式。

\frac{\sin (x) (1 + \cos (x))}{(1 + \cos (x)) (1 - \cos (x))} + \frac{\sin (x) (1 - \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}

$\frac{\sin (x) (1 + \cos (x))}{(1 + \cos (x)) (1 - \cos (x))} + \frac{\sin (x) (1 - \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}$

\frac{\sin (x) (1 + \cos (x))}{(1 + \cos (x)) (1 - \cos (x))} + \frac{\sin (x) (1 - \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}

$\frac{\sin (x) (1 + \cos (x))}{(1 + \cos (x)) (1 - \cos (x))} + \frac{\sin (x) (1 - \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}$

解题步骤 2.4

在公分母上合并分子。

\frac{\sin (x) (1 + \cos (x)) + \sin (x) (1 - \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}

$\frac{\sin (x) (1 + \cos (x)) + \sin (x) (1 - \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}$

\frac{\sin (x) (1 + \cos (x)) + \sin (x) (1 - \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}

$\frac{\sin (x) (1 + \cos (x)) + \sin (x) (1 - \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}$

解题步骤 3

化简分子。

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解题步骤 3.1

化简每一项。

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解题步骤 3.1.1

运用分配律。

\frac{\sin (x) \cdot 1 + \sin (x) \cos (x) + \sin (x) (1 - \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}

$\frac{\sin (x) \cdot 1 + \sin (x) \cos (x) + \sin (x) (1 - \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}$

解题步骤 3.1.2

将

\sin (x)

$\sin (x)$ 乘以

1

$1$ 。

\frac{\sin (x) + \sin (x) \cos (x) + \sin (x) (1 - \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}

$\frac{\sin (x) + \sin (x) \cos (x) + \sin (x) (1 - \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}$

解题步骤 3.1.3

运用分配律。

\frac{\sin (x) + \sin (x) \cos (x) + \sin (x) \cdot 1 + \sin (x) (- \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}

$\frac{\sin (x) + \sin (x) \cos (x) + \sin (x) \cdot 1 + \sin (x) (- \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}$

解题步骤 3.1.4

将

\sin (x)

$\sin (x)$ 乘以

1

$1$ 。

\frac{\sin (x) + \sin (x) \cos (x) + \sin (x) + \sin (x) (- \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}

$\frac{\sin (x) + \sin (x) \cos (x) + \sin (x) + \sin (x) (- \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}$

\frac{\sin (x) + \sin (x) \cos (x) + \sin (x) + \sin (x) (- \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}

$\frac{\sin (x) + \sin (x) \cos (x) + \sin (x) + \sin (x) (- \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}$

解题步骤 3.2

将

\sin (x)

$\sin (x)$ 和

\sin (x)

$\sin (x)$ 相加。

\frac{\sin (x) \cos (x) + 2 \sin (x) + \sin (x) (- \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}

$\frac{\sin (x) \cos (x) + 2 \sin (x) + \sin (x) (- \cos (x))}{(1 + \cos (x)) (1 - \cos (x))}$

解题步骤 3.3

将

\sin (x) \cos (x)

$\sin (x) \cos (x)$ 和

\sin (x) (- \cos (x))

$\sin (x) (- \cos (x))$ 相加。

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解题步骤 3.3.1

将

\sin (x)

$\sin (x)$ 和

- 1

$- 1$ 重新排序。

\frac{2 \sin (x) + \sin (x) \cos (x) - 1 \cdot \sin (x) \cos (x)}{(1 + \cos (x)) (1 - \cos (x))}

$\frac{2 \sin (x) + \sin (x) \cos (x) - 1 \cdot \sin (x) \cos (x)}{(1 + \cos (x)) (1 - \cos (x))}$

解题步骤 3.3.2

从

\sin (x) \cos (x)

$\sin (x) \cos (x)$ 中减去

\sin (x) \cos (x)

$\sin (x) \cos (x)$ 。

\frac{2 \sin (x) + 0}{(1 + \cos (x)) (1 - \cos (x))}

$\frac{2 \sin (x) + 0}{(1 + \cos (x)) (1 - \cos (x))}$

\frac{2 \sin (x) + 0}{(1 + \cos (x)) (1 - \cos (x))}

$\frac{2 \sin (x) + 0}{(1 + \cos (x)) (1 - \cos (x))}$

解题步骤 3.4

将

2 \sin (x)

$2 \sin (x)$ 和

0

$0$ 相加。

\frac{2 \sin (x)}{(1 + \cos (x)) (1 - \cos (x))}

$\frac{2 \sin (x)}{(1 + \cos (x)) (1 - \cos (x))}$

\frac{2 \sin (x)}{(1 + \cos (x)) (1 - \cos (x))}

$\frac{2 \sin (x)}{(1 + \cos (x)) (1 - \cos (x))}$

解题步骤 4

化简分母。

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解题步骤 4.1

使用 FOIL 方法展开

(1 + \cos (x)) (1 - \cos (x))

$(1 + \cos (x)) (1 - \cos (x))$ 。

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解题步骤 4.1.1

运用分配律。

\frac{2 \sin (x)}{1 (1 - \cos (x)) + \cos (x) (1 - \cos (x))}

$\frac{2 \sin (x)}{1 (1 - \cos (x)) + \cos (x) (1 - \cos (x))}$

解题步骤 4.1.2

运用分配律。

\frac{2 \sin (x)}{1 \cdot 1 + 1 (- \cos (x)) + \cos (x) (1 - \cos (x))}

$\frac{2 \sin (x)}{1 \cdot 1 + 1 (- \cos (x)) + \cos (x) (1 - \cos (x))}$

解题步骤 4.1.3

运用分配律。

\frac{2 \sin (x)}{1 \cdot 1 + 1 (- \cos (x)) + \cos (x) \cdot 1 + \cos (x) (- \cos (x))}

$\frac{2 \sin (x)}{1 \cdot 1 + 1 (- \cos (x)) + \cos (x) \cdot 1 + \cos (x) (- \cos (x))}$

\frac{2 \sin (x)}{1 \cdot 1 + 1 (- \cos (x)) + \cos (x) \cdot 1 + \cos (x) (- \cos (x))}

$\frac{2 \sin (x)}{1 \cdot 1 + 1 (- \cos (x)) + \cos (x) \cdot 1 + \cos (x) (- \cos (x))}$

解题步骤 4.2

化简并合并同类项。

\frac{2 \sin (x)}{1 - \cos^{2} (x)}

$\frac{2 \sin (x)}{1 - \cos^{2} (x)}$

\frac{2 \sin (x)}{1 - \cos^{2} (x)}

$\frac{2 \sin (x)}{1 - \cos^{2} (x)}$

解题步骤 5

使用勾股恒等式。

\frac{2 \sin (x)}{\sin^{2} (x)}

$\frac{2 \sin (x)}{\sin^{2} (x)}$

解题步骤 6

约去

\sin (x)

$\sin (x)$ 和

{\sin (x)}^{2}

${\sin (x)}^{2}$ 的公因数。

\frac{2}{\sin (x)}

$\frac{2}{\sin (x)}$

解题步骤 7

将

\frac{2}{\sin (x)}

$\frac{2}{\sin (x)}$ 重写为

2 \csc (x)

$2 \csc (x)$ 。

2 \csc (x)

$2 \csc (x)$

解题步骤 8

因为两边已证明为相等，所以方程为恒等式。

\frac{\sin (x)}{1 - \cos (x)} + \frac{\sin (x)}{1 + \cos (x)} = 2 \csc (x)

$\frac{\sin (x)}{1 - \cos (x)} + \frac{\sin (x)}{1 + \cos (x)} = 2 \csc (x)$ 是一个恒等式

三角学 示例

三角学示例