三角学示例

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

解题步骤 1

化简每一项。

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解题步骤 1.1

将 $\tan (x)$ 重写为正弦和余弦形式。

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

解题步骤 1.2

将 $\cot (x)$ 重写为正弦和余弦形式。

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

解题步骤 2

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

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

解题步骤 3

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

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

解题步骤 4

通过与 $1$ 的合适因数相乘，将每一个表达式写成具有公分母 $(1 - \frac{\sin (x)}{\cos (x)}) (1 - \frac{\cos (x)}{\sin (x)})$ 的形式。

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解题步骤 4.1

将 $\frac{\cos (x)}{1 - \frac{\sin (x)}{\cos (x)}}$ 乘以 $\frac{1 - \frac{\cos (x)}{\sin (x)}}{1 - \frac{\cos (x)}{\sin (x)}}$ 。

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

解题步骤 4.2

将 $\frac{\sin (x)}{1 - \frac{\cos (x)}{\sin (x)}}$ 乘以 $\frac{1 - \frac{\sin (x)}{\cos (x)}}{1 - \frac{\sin (x)}{\cos (x)}}$ 。

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

解题步骤 4.3

重新排序 $(1 - \frac{\sin (x)}{\cos (x)}) (1 - \frac{\cos (x)}{\sin (x)})$ 的因式。

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

解题步骤 5

在公分母上合并分子。

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

解题步骤 6

化简分子。

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解题步骤 6.1

运用分配律。

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

解题步骤 6.2

将 $\cos (x)$ 乘以 $1$ 。

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

解题步骤 6.3

使用乘法的交换性质重写。

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

解题步骤 6.4

乘以 $- \cos (x) \frac{\cos (x)}{\sin (x)}$ 。

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解题步骤 6.4.1

组合 $\frac{\cos (x)}{\sin (x)}$ 和 $\cos (x)$ 。

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

解题步骤 6.4.2

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 6.4.3

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 6.4.4

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 6.4.5

将 $1$ 和 $1$ 相加。

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

解题步骤 6.5

运用分配律。

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

解题步骤 6.6

将 $\sin (x)$ 乘以 $1$ 。

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

解题步骤 6.7

使用乘法的交换性质重写。

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

解题步骤 6.8

乘以 $- \sin (x) \frac{\sin (x)}{\cos (x)}$ 。

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解题步骤 6.8.1

组合 $\frac{\sin (x)}{\cos (x)}$ 和 $\sin (x)$ 。

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

解题步骤 6.8.2

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 6.8.3

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 6.8.4

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 6.8.5

将 $1$ 和 $1$ 相加。

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

解题步骤 7

化简分子。

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解题步骤 7.1

从 $\cos^{2} (x)$ 中分解出因数 $\cos (x)$ 。

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

解题步骤 7.2

分离分数。

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

解题步骤 7.3

将 $\frac{\cos (x)}{\sin (x)}$ 转换成 $\cot (x)$ 。

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

解题步骤 7.4

用 $\cos (x)$ 除以 $1$ 。

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

解题步骤 7.5

从 $\sin^{2} (x)$ 中分解出因数 $\sin (x)$ 。

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

解题步骤 7.6

分离分数。

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

解题步骤 7.7

将 $\frac{\sin (x)}{\cos (x)}$ 转换成 $\tan (x)$ 。

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

解题步骤 7.8

用 $\sin (x)$ 除以 $1$ 。

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

解题步骤 8

化简分母。

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解题步骤 8.1

将 $\frac{\cos (x)}{\sin (x)}$ 转换成 $\cot (x)$ 。

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

解题步骤 8.2

将 $\frac{\sin (x)}{\cos (x)}$ 转换成 $\tan (x)$ 。

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

三角学 示例

三角学示例