三角学示例

$\frac{\sec^{4} (x) - \tan^{4} (x)}{\sec^{2} (x) + \tan^{2} (x)} = \sec^{2} (x) - \tan^{2} (x)$

解题步骤 1

从左边开始。

$\frac{\sec^{4} (x) - \tan^{4} (x)}{\sec^{2} (x) + \tan^{2} (x)}$

解题步骤 2

转换成正弦和余弦。

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解题步骤 2.1

对 $\sec (x)$ 使用倒数恒等式。

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

解题步骤 2.2

使用商数恒等式以正弦和余弦书写 $\tan (x)$ 。

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

解题步骤 2.3

对 $\sec (x)$ 使用倒数恒等式。

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

解题步骤 2.4

使用商数恒等式以正弦和余弦书写 $\tan (x)$ 。

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

解题步骤 2.5

对 $\frac{1}{\cos (x)}$ 运用乘积法则。

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

解题步骤 2.6

对 $\frac{\sin (x)}{\cos (x)}$ 运用乘积法则。

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

解题步骤 2.7

对 $\frac{1}{\cos (x)}$ 运用乘积法则。

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

解题步骤 2.8

对 $\frac{\sin (x)}{\cos (x)}$ 运用乘积法则。

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

解题步骤 3

化简。

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解题步骤 3.1

Multiply the numerator and denominator of the fraction by ${\cos (x)}^{4}$ .

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解题步骤 3.1.1

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

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

解题步骤 3.1.2

合并。

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

解题步骤 3.2

运用分配律。

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

解题步骤 3.3

通过相约进行化简。

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解题步骤 3.3.1

约去 ${\cos (x)}^{4}$ 的公因数。

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解题步骤 3.3.1.1

约去公因数。

解题步骤 3.3.1.2

重写表达式。

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

解题步骤 3.3.2

约去 ${\cos (x)}^{4}$ 的公因数。

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解题步骤 3.3.2.1

将 $- \frac{{\sin (x)}^{4}}{{\cos (x)}^{4}}$ 中前置负号移到分子中。

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

解题步骤 3.3.2.2

约去公因数。

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

解题步骤 3.3.2.3

重写表达式。

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

解题步骤 3.3.3

约去 ${\cos (x)}^{2}$ 的公因数。

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解题步骤 3.3.3.1

从 ${\cos (x)}^{4}$ 中分解出因数 ${\cos (x)}^{2}$ 。

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

解题步骤 3.3.3.2

约去公因数。

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

解题步骤 3.3.3.3

重写表达式。

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

解题步骤 3.3.4

约去 ${\cos (x)}^{2}$ 的公因数。

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解题步骤 3.3.4.1

从 ${\cos (x)}^{4}$ 中分解出因数 ${\cos (x)}^{2}$ 。

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

解题步骤 3.3.4.2

约去公因数。

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

解题步骤 3.3.4.3

重写表达式。

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

解题步骤 3.4

化简分子。

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解题步骤 3.4.1

将 $1^{4}$ 重写为 ${(1^{2})}^{2}$ 。

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

解题步骤 3.4.2

将 ${\sin (x)}^{4}$ 重写为 ${({\sin (x)}^{2})}^{2}$ 。

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

解题步骤 3.4.3

因为两项都是完全平方数，所以使用平方差公式 $a^{2} - b^{2} = (a + b) (a - b)$ 进行因式分解，其中 $a = 1^{2}$ 和 $b = {\sin (x)}^{2}$ 。

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

解题步骤 3.4.4

化简。

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解题步骤 3.4.4.1

一的任意次幂都为一。

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

解题步骤 3.4.4.2

因为两项都是完全平方数，所以使用平方差公式 $a^{2} - b^{2} = (a + b) (a - b)$ 进行因式分解，其中 $a = 1$ 和 $b = \sin (x)$ 。

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

解题步骤 3.5

化简分母。

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解题步骤 3.5.1

从 ${\cos (x)}^{2} \cdot 1^{2} + {\cos (x)}^{2} {\sin (x)}^{2}$ 中分解出因数 ${\cos (x)}^{2}$ 。

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解题步骤 3.5.1.1

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

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

解题步骤 3.5.1.2

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

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

解题步骤 3.5.1.3

从 ${\cos (x)}^{2} (1^{2}) + {\cos (x)}^{2} ({\sin (x)}^{2})$ 中分解出因数 ${\cos (x)}^{2}$ 。

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

解题步骤 3.5.2

一的任意次幂都为一。

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

解题步骤 3.6

约去 $1 + {\sin (x)}^{2}$ 的公因数。

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

解题步骤 4

现在，考虑等式的右边。

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

解题步骤 5

转换成正弦和余弦。

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解题步骤 5.1

对 $\sec (x)$ 使用倒数恒等式。

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

解题步骤 5.2

使用商数恒等式以正弦和余弦书写 $\tan (x)$ 。

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

解题步骤 5.3

对 $\frac{1}{\cos (x)}$ 运用乘积法则。

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

解题步骤 5.4

对 $\frac{\sin (x)}{\cos (x)}$ 运用乘积法则。

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

解题步骤 6

一的任意次幂都为一。

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

解题步骤 7

在公分母上合并分子。

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

解题步骤 8

化简分子。

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

解题步骤 9

因为两边已证明为相等，所以方程为恒等式。

$\frac{\sec^{4} (x) - \tan^{4} (x)}{\sec^{2} (x) + \tan^{2} (x)} = \sec^{2} (x) - \tan^{2} (x)$ 是一个恒等式

三角学 示例

三角学示例