三角学示例

$\sec^{2} (x) + \cot^{2} (x) = \tan^{2} (x) + \csc^{2} (x)$

解题步骤 1

从左边开始。

$\sec^{2} (x) + \cot^{2} (x)$

解题步骤 2

将勾股恒等式反过来使用。

$\tan^{2} (x) + 1 + \cot^{2} (x)$

解题步骤 3

转换成正弦和余弦。

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解题步骤 3.1

使用商数恒等式以正弦和余弦书写 $\tan (x)$ 。

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

解题步骤 3.2

使用商数恒等式以正弦和余弦书写 $\cot (x)$ 。

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

解题步骤 3.3

对 $\frac{\sin (x)}{\cos (x)}$ 运用乘积法则。

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

解题步骤 3.4

对 $\frac{\cos (x)}{\sin (x)}$ 运用乘积法则。

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

解题步骤 4

将 $\frac{\sin^{2} (x)}{\cos^{2} (x)} + 1$ 写成分母为 $1$ 的分数。

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

解题步骤 5

加上分数。

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解题步骤 5.1

要将 $\frac{\frac{\sin^{2} (x)}{\cos^{2} (x)} + 1}{1}$ 写成带有公分母的分数，请乘以 $\frac{\sin^{2} (x)}{\sin^{2} (x)}$ 。

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

解题步骤 5.2

将 $\frac{\frac{\sin^{2} (x)}{\cos^{2} (x)} + 1}{1}$ 乘以 $\frac{\sin^{2} (x)}{\sin^{2} (x)}$ 。

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

解题步骤 5.3

在公分母上合并分子。

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

解题步骤 6

化简每一项。

$\frac{\frac{\sin^{4} (x)}{\cos^{2} (x)} + \sin^{2} (x) + \cos^{2} (x)}{\sin^{2} (x)}$

解题步骤 7

将 $\sin^{2} (x)$ 写成分母为 $1$ 的分数。

$\frac{\frac{\sin^{4} (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{1} + \cos^{2} (x)}{\sin^{2} (x)}$

解题步骤 8

加上分数。

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解题步骤 8.1

要将 $\frac{\sin^{2} (x)}{1}$ 写成带有公分母的分数，请乘以 $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ 。

$\frac{\frac{\sin^{4} (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{1} \cdot \frac{\cos^{2} (x)}{\cos^{2} (x)} + \cos^{2} (x)}{\sin^{2} (x)}$

解题步骤 8.2

将 $\frac{\sin^{2} (x)}{1}$ 乘以 $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ 。

$\frac{\frac{\sin^{4} (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x) \cos^{2} (x)}{\cos^{2} (x)} + \cos^{2} (x)}{\sin^{2} (x)}$

解题步骤 8.3

在公分母上合并分子。

$\frac{\frac{\sin^{4} (x) + \sin^{2} (x) \cos^{2} (x)}{\cos^{2} (x)} + \cos^{2} (x)}{\sin^{2} (x)}$

解题步骤 9

将 $\cos^{2} (x)$ 写成分母为 $1$ 的分数。

$\frac{\frac{\sin^{4} (x) + \sin^{2} (x) \cos^{2} (x)}{\cos^{2} (x)} + \frac{\cos^{2} (x)}{1}}{\sin^{2} (x)}$

解题步骤 10

加上分数。

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解题步骤 10.1

要将 $\frac{\cos^{2} (x)}{1}$ 写成带有公分母的分数，请乘以 $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ 。

$\frac{\frac{\sin^{4} (x) + \sin^{2} (x) \cos^{2} (x)}{\cos^{2} (x)} + \frac{\cos^{2} (x)}{1} \cdot \frac{\cos^{2} (x)}{\cos^{2} (x)}}{\sin^{2} (x)}$

解题步骤 10.2

将 $\frac{\cos^{2} (x)}{1}$ 乘以 $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ 。

$\frac{\frac{\sin^{4} (x) + \sin^{2} (x) \cos^{2} (x)}{\cos^{2} (x)} + \frac{\cos^{2} (x) \cos^{2} (x)}{\cos^{2} (x)}}{\sin^{2} (x)}$

解题步骤 10.3

在公分母上合并分子。

$\frac{\frac{\sin^{4} (x) + \sin^{2} (x) \cos^{2} (x) + \cos^{2} (x) \cos^{2} (x)}{\cos^{2} (x)}}{\sin^{2} (x)}$

解题步骤 11

通过指数相加将 ${\cos (x)}^{2}$ 乘以 ${\cos (x)}^{2}$ 。

$\frac{\frac{\sin^{4} (x) + \sin^{2} (x) \cos^{2} (x) + \cos^{4} (x)}{\cos^{2} (x)}}{\sin^{2} (x)}$

解题步骤 12

将勾股恒等式反过来使用。

$\frac{\frac{\sin^{4} (x) + (1 - \cos^{2} (x)) \cos^{2} (x) + \cos^{4} (x)}{\cos^{2} (x)}}{\sin^{2} (x)}$

解题步骤 13

化简。

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解题步骤 13.1

将分子乘以分母的倒数。

$\frac{{\sin (x)}^{4} + (1 - {\cos (x)}^{2}) {\cos (x)}^{2} + {\cos (x)}^{4}}{{\cos (x)}^{2}} \cdot \frac{1}{{\sin (x)}^{2}}$

解题步骤 13.2

化简分子。

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解题步骤 13.2.1

运用分配律。

$\frac{{\sin (x)}^{4} + 1 {\cos (x)}^{2} - {\cos (x)}^{2} {\cos (x)}^{2} + {\cos (x)}^{4}}{{\cos (x)}^{2}} \cdot \frac{1}{{\sin (x)}^{2}}$

解题步骤 13.2.2

将 ${\cos (x)}^{2}$ 乘以 $1$ 。

$\frac{{\sin (x)}^{4} + {\cos (x)}^{2} - {\cos (x)}^{2} {\cos (x)}^{2} + {\cos (x)}^{4}}{{\cos (x)}^{2}} \cdot \frac{1}{{\sin (x)}^{2}}$

解题步骤 13.2.3

通过指数相加将 ${\cos (x)}^{2}$ 乘以 ${\cos (x)}^{2}$ 。

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解题步骤 13.2.3.1

移动 ${\cos (x)}^{2}$ 。

$\frac{{\sin (x)}^{4} + {\cos (x)}^{2} - ({\cos (x)}^{2} {\cos (x)}^{2}) + {\cos (x)}^{4}}{{\cos (x)}^{2}} \cdot \frac{1}{{\sin (x)}^{2}}$

解题步骤 13.2.3.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$\frac{{\sin (x)}^{4} + {\cos (x)}^{2} - {\cos (x)}^{2 + 2} + {\cos (x)}^{4}}{{\cos (x)}^{2}} \cdot \frac{1}{{\sin (x)}^{2}}$

解题步骤 13.2.3.3

将 $2$ 和 $2$ 相加。

$\frac{{\sin (x)}^{4} + {\cos (x)}^{2} - {\cos (x)}^{4} + {\cos (x)}^{4}}{{\cos (x)}^{2}} \cdot \frac{1}{{\sin (x)}^{2}}$

解题步骤 13.2.4

将 $- {\cos (x)}^{4}$ 和 ${\cos (x)}^{4}$ 相加。

$\frac{{\sin (x)}^{4} + {\cos (x)}^{2} + 0}{{\cos (x)}^{2}} \cdot \frac{1}{{\sin (x)}^{2}}$

解题步骤 13.2.5

将 ${\sin (x)}^{4} + {\cos (x)}^{2}$ 和 $0$ 相加。

$\frac{{\sin (x)}^{4} + {\cos (x)}^{2}}{{\cos (x)}^{2}} \cdot \frac{1}{{\sin (x)}^{2}}$

解题步骤 13.3

将 $\frac{{\sin (x)}^{4} + {\cos (x)}^{2}}{{\cos (x)}^{2}}$ 乘以 $\frac{1}{{\sin (x)}^{2}}$ 。

$\frac{\sin^{4} (x) + \cos^{2} (x)}{\cos^{2} (x) \sin^{2} (x)}$

解题步骤 14

现在，考虑等式的右边。

$\tan^{2} (x) + \csc^{2} (x)$

解题步骤 15

转换成正弦和余弦。

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解题步骤 15.1

使用商数恒等式以正弦和余弦书写 $\tan (x)$ 。

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

解题步骤 15.2

对 $\csc (x)$ 使用倒数恒等式。

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

解题步骤 15.3

对 $\frac{\sin (x)}{\cos (x)}$ 运用乘积法则。

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

解题步骤 15.4

对 $\frac{1}{\sin (x)}$ 运用乘积法则。

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

解题步骤 16

一的任意次幂都为一。

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

解题步骤 17

加上分数。

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解题步骤 17.1

要将 $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ 写成带有公分母的分数，请乘以 $\frac{\sin^{2} (x)}{\sin^{2} (x)}$ 。

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

解题步骤 17.2

要将 $\frac{1}{\sin^{2} (x)}$ 写成带有公分母的分数，请乘以 $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ 。

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

解题步骤 17.3

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 $\cos^{2} (x) \sin^{2} (x)$ 的形式。

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解题步骤 17.3.1

将 $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ 乘以 $\frac{\sin^{2} (x)}{\sin^{2} (x)}$ 。

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

解题步骤 17.3.2

将 $\frac{1}{\sin^{2} (x)}$ 乘以 $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ 。

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

解题步骤 17.3.3

重新排序 $\sin^{2} (x) \cos^{2} (x)$ 的因式。

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

解题步骤 17.4

在公分母上合并分子。

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

解题步骤 18

通过指数相加将 ${\sin (x)}^{2}$ 乘以 ${\sin (x)}^{2}$ 。

$\frac{\sin^{4} (x) + \cos^{2} (x)}{\cos^{2} (x) \sin^{2} (x)}$

解题步骤 19

因为两边已证明为相等，所以方程为恒等式。

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

三角学 示例

三角学示例