三角学示例

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

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

解题步骤 1

从左边开始。

\frac{{csc}^{2} (x)}{{cot}^{2} (x) - 1}

$\frac{\csc^{2} (x)}{\cot^{2} (x) - 1}$

解题步骤 2

转换成正弦和余弦。

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解题步骤 2.1

对

csc (x)

$\csc (x)$ 使用倒数恒等式。

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

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

解题步骤 2.2

使用商数恒等式以正弦和余弦书写

cot (x)

$\cot (x)$ 。

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

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

解题步骤 2.3

对

\frac{1}{sin (x)}

$\frac{1}{\sin (x)}$ 运用乘积法则。

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

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

解题步骤 2.4

对

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

$\frac{\cos (x)}{\sin (x)}$ 运用乘积法则。

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

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

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

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

解题步骤 3

化简。

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解题步骤 3.1

将分子乘以分母的倒数。

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

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

解题步骤 3.2

一的任意次幂都为一。

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

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

解题步骤 3.3

化简分母。

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解题步骤 3.3.1

将

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

$\frac{{\cos (x)}^{2}}{{\sin (x)}^{2}}$ 重写为

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

${(\frac{\cos (x)}{\sin (x)})}^{2}$ 。

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

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

解题步骤 3.3.2

将

1

$1$ 重写为

1^{2}

$1^{2}$ 。

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

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

解题步骤 3.3.3

因为两项都是完全平方数，所以使用平方差公式

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ 进行因式分解，其中

a = \frac{cos (x)}{sin (x)}

$a = \frac{\cos (x)}{\sin (x)}$ 和

b = 1

$b = 1$ 。

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

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

解题步骤 3.3.4

将

1

$1$ 写成具有公分母的分数。

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

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

解题步骤 3.3.5

在公分母上合并分子。

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

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

解题步骤 3.3.6

要将

- 1

$- 1$ 写成带有公分母的分数，请乘以

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

$\frac{\sin (x)}{\sin (x)}$ 。

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

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

解题步骤 3.3.7

组合

- 1

$- 1$ 和

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

$\frac{\sin (x)}{\sin (x)}$ 。

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

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

解题步骤 3.3.8

在公分母上合并分子。

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

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

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

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

解题步骤 3.4

将

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

$\frac{\cos (x) + \sin (x)}{\sin (x)}$ 乘以

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

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

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

解题步骤 3.5

化简分母。

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解题步骤 3.5.1

对

$\sin (x)$ 进行

$1$ 次方运算。

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

解题步骤 3.5.2

对

$\sin (x)$ 进行

$1$ 次方运算。

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

解题步骤 3.5.3

使用幂法则

$a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 3.5.4

将

$1$ 和

$1$ 相加。

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

解题步骤 3.6

合并。

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

解题步骤 3.7

将

$1$ 乘以

$1$ 。

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

解题步骤 3.8

组合

${\sin (x)}^{2}$ 和

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

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

解题步骤 3.9

通过约去公因数来化简表达式。

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

解题步骤 4

将

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

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

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

解题步骤 5

因为两边已证明为相等，所以方程为恒等式。

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

三角学 示例

三角学示例