三角学示例

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x)}{1 - \sec (x)} = 2 \cot (x)$

解题步骤 1

从左边开始。

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x)}{1 - \sec (x)}$

解题步骤 2

将 $\frac{\tan (x)}{1 - \sec (x)}$ 乘以 $\frac{1 + \sec (x)}{1 + \sec (x)}$ 。

$\frac{1 - \sec (x)}{\tan (x)} - (\frac{\tan (x)}{1 - \sec (x)} \cdot \frac{1 + \sec (x)}{1 + \sec (x)})$

解题步骤 3

合并。

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x) (1 + \sec (x))}{(1 - \sec (x)) (1 + \sec (x))}$

解题步骤 4

化简分子。

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解题步骤 4.1

运用分配律。

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x) \cdot 1 + \tan (x) \sec (x)}{(1 - \sec (x)) (1 + \sec (x))}$

解题步骤 4.2

将 $\tan (x)$ 乘以 $1$ 。

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x) + \tan (x) \sec (x)}{(1 - \sec (x)) (1 + \sec (x))}$

解题步骤 5

化简分母。

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解题步骤 5.1

使用 FOIL 方法展开 $(1 - \sec (x)) (1 + \sec (x))$ 。

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解题步骤 5.1.1

运用分配律。

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x) + \tan (x) \sec (x)}{1 (1 + \sec (x)) - \sec (x) (1 + \sec (x))}$

解题步骤 5.1.2

运用分配律。

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x) + \tan (x) \sec (x)}{1 \cdot 1 + 1 \sec (x) - \sec (x) (1 + \sec (x))}$

解题步骤 5.1.3

运用分配律。

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x) + \tan (x) \sec (x)}{1 \cdot 1 + 1 \sec (x) - \sec (x) \cdot 1 - \sec (x) \sec (x)}$

解题步骤 5.2

化简并合并同类项。

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x) + \tan (x) \sec (x)}{1 - \sec^{2} (x)}$

解题步骤 6

将 $- 1$ 写成分母为 $1$ 的分数。

$\frac{1 - \sec (x)}{\tan (x)} + \frac{- 1}{1} \cdot \frac{\tan (x) + \tan (x) \sec (x)}{1 - \sec^{2} (x)}$

解题步骤 7

合并。

$\frac{1 - \sec (x)}{\tan (x)} + \frac{- (\tan (x) + \tan (x) \sec (x))}{1 (1 - \sec^{2} (x))}$

解题步骤 8

运用分配律。

$\frac{1 - \sec (x)}{\tan (x)} + \frac{- \tan (x) - \tan (x) \sec (x)}{1 (1 - \sec^{2} (x))}$

解题步骤 9

将 $1 - {\sec (x)}^{2}$ 乘以 $1$ 。

$\frac{1 - \sec (x)}{\tan (x)} + \frac{- \tan (x) - \tan (x) \sec (x)}{1 - \sec^{2} (x)}$

解题步骤 10

使用勾股恒等式。

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解题步骤 10.1

将 $1$ 和 $- \sec^{2} (x)$ 重新排序。

$\frac{1 - \sec (x)}{\tan (x)} + \frac{- \tan (x) - \tan (x) \sec (x)}{- \sec^{2} (x) + 1}$

解题步骤 10.2

从 $- \sec^{2} (x)$ 中分解出因数 $- 1$ 。

$\frac{1 - \sec (x)}{\tan (x)} + \frac{- \tan (x) - \tan (x) \sec (x)}{- (\sec^{2} (x)) + 1}$

解题步骤 10.3

将 $1$ 重写为 $- 1 (- 1)$ 。

$\frac{1 - \sec (x)}{\tan (x)} + \frac{- \tan (x) - \tan (x) \sec (x)}{- \sec^{2} (x) - 1 \cdot - 1}$

解题步骤 10.4

从 $- \sec^{2} (x) - 1 \cdot - 1$ 中分解出因数 $- 1$ 。

$\frac{1 - \sec (x)}{\tan (x)} + \frac{- \tan (x) - \tan (x) \sec (x)}{- (\sec^{2} (x) - 1)}$

解题步骤 10.5

使用勾股恒等式。

$\frac{1 - \sec (x)}{\tan (x)} + \frac{- \tan (x) - \tan (x) \sec (x)}{- \tan^{2} (x)}$

解题步骤 11

转换成正弦和余弦。

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解题步骤 11.1

对 $\sec (x)$ 使用倒数恒等式。

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

解题步骤 11.2

使用商数恒等式以正弦和余弦书写 $\tan (x)$ 。

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

解题步骤 11.3

使用商数恒等式以正弦和余弦书写 $\tan (x)$ 。

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

解题步骤 11.4

使用商数恒等式以正弦和余弦书写 $\tan (x)$ 。

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

解题步骤 11.5

对 $\sec (x)$ 使用倒数恒等式。

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

解题步骤 11.6

使用商数恒等式以正弦和余弦书写 $\tan (x)$ 。

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

解题步骤 11.7

对 $\frac{\sin (x)}{\cos (x)}$ 运用乘积法则。

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

解题步骤 12

化简。

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解题步骤 12.1

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

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

解题步骤 12.2

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

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

解题步骤 12.3

组合 $- \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ 和 $\frac{\cos (x)}{\cos (x)}$ 。

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

解题步骤 12.4

在公分母上合并分子。

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

解题步骤 12.5

化简每一项。

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解题步骤 12.5.1

将分子乘以分母的倒数。

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

解题步骤 12.5.2

运用分配律。

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

解题步骤 12.5.3

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

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

解题步骤 12.5.4

约去 $\cos (x)$ 的公因数。

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解题步骤 12.5.4.1

将 $- \frac{1}{\cos (x)}$ 中前置负号移到分子中。

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

解题步骤 12.5.4.2

约去公因数。

解题步骤 12.5.4.3

重写表达式。

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

解题步骤 12.5.5

将 $- 1 \frac{1}{\sin (x)}$ 重写为 $- \frac{1}{\sin (x)}$ 。

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

解题步骤 12.5.6

将分子乘以分母的倒数。

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

解题步骤 12.5.7

约去 $\cos (x)$ 的公因数。

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解题步骤 12.5.7.1

将 $- \frac{{\cos (x)}^{2}}{{\sin (x)}^{2}}$ 中前置负号移到分子中。

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

解题步骤 12.5.7.2

从 $- {\cos (x)}^{2}$ 中分解出因数 $\cos (x)$ 。

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

解题步骤 12.5.7.3

约去公因数。

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

解题步骤 12.5.7.4

重写表达式。

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

解题步骤 12.5.8

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

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

解题步骤 12.5.9

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 12.5.10

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 12.5.11

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 12.5.12

将 $1$ 和 $1$ 相加。

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

解题步骤 12.5.13

组合 $\cos (x)$ 和 $\frac{\sin (x)}{{\cos (x)}^{2}}$ 。

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

解题步骤 12.5.14

约去 $\cos (x)$ 和 ${\cos (x)}^{2}$ 的公因数。

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解题步骤 12.5.14.1

从 $\cos (x) \sin (x)$ 中分解出因数 $\cos (x)$ 。

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

解题步骤 12.5.14.2

约去公因数。

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解题步骤 12.5.14.2.1

从 ${\cos (x)}^{2}$ 中分解出因数 $\cos (x)$ 。

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

解题步骤 12.5.14.2.2

约去公因数。

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

解题步骤 12.5.14.2.3

重写表达式。

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

解题步骤 12.5.15

将负号移到分数的前面。

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

解题步骤 12.5.16

运用分配律。

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

解题步骤 12.5.17

约去 $\sin (x)$ 的公因数。

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解题步骤 12.5.17.1

将 $- \frac{\cos (x)}{{\sin (x)}^{2}}$ 中前置负号移到分子中。

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

解题步骤 12.5.17.2

从 $- \sin (x)$ 中分解出因数 $\sin (x)$ 。

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

解题步骤 12.5.17.3

从 ${\sin (x)}^{2}$ 中分解出因数 $\sin (x)$ 。

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

解题步骤 12.5.17.4

约去公因数。

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

解题步骤 12.5.17.5

重写表达式。

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

解题步骤 12.5.18

约去 $\sin (x)$ 的公因数。

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解题步骤 12.5.18.1

将 $- \frac{\sin (x)}{\cos (x)}$ 中前置负号移到分子中。

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

解题步骤 12.5.18.2

将 $- \frac{\cos (x)}{{\sin (x)}^{2}}$ 中前置负号移到分子中。

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

解题步骤 12.5.18.3

从 $- \sin (x)$ 中分解出因数 $\sin (x)$ 。

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

解题步骤 12.5.18.4

从 ${\sin (x)}^{2}$ 中分解出因数 $\sin (x)$ 。

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

解题步骤 12.5.18.5

约去公因数。

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

解题步骤 12.5.18.6

重写表达式。

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

解题步骤 12.5.19

约去 $\cos (x)$ 的公因数。

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解题步骤 12.5.19.1

从 $- \cos (x)$ 中分解出因数 $\cos (x)$ 。

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

解题步骤 12.5.19.2

约去公因数。

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

解题步骤 12.5.19.3

重写表达式。

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

解题步骤 12.5.20

化简每一项。

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解题步骤 12.5.20.1

将负号移到分数的前面。

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

解题步骤 12.5.20.2

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

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解题步骤 12.5.20.2.1

将 $- 1$ 乘以 $- 1$ 。

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

解题步骤 12.5.20.2.2

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

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

解题步骤 12.5.20.3

将负号移到分数的前面。

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

解题步骤 12.5.20.4

乘以 $- 1 (- \frac{1}{\sin (x)})$ 。

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解题步骤 12.5.20.4.1

将 $- 1$ 乘以 $- 1$ 。

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

解题步骤 12.5.20.4.2

将 $\frac{1}{\sin (x)}$ 乘以 $1$ 。

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

解题步骤 12.6

在公分母上合并分子。

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

解题步骤 12.7

将 $\cos (x)$ 和 $\cos (x)$ 相加。

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

解题步骤 12.8

将 $- 1$ 和 $1$ 相加。

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

解题步骤 12.9

将 $2 \cos (x)$ 和 $0$ 相加。

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

解题步骤 13

将 $\frac{2 \cos (x)}{\sin (x)}$ 重写为 $2 \cot (x)$ 。

$2 \cot (x)$

解题步骤 14

因为两边已证明为相等，所以方程为恒等式。

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x)}{1 - \sec (x)} = 2 \cot (x)$ 是一个恒等式

三角学 示例

三角学示例