初级微积分示例

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

解题步骤 1

从左边开始。

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

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。

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

解题步骤 2.2

使用商数恒等式以正弦和余弦书写 $\tan (x)$ 。

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

解题步骤 3

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

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

解题步骤 4

合并。

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

解题步骤 5

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

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

解题步骤 6

化简分母。

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解题步骤 6.1

使用 FOIL 方法展开 $(\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)})$ 。

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解题步骤 6.1.1

运用分配律。

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

解题步骤 6.1.2

运用分配律。

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

解题步骤 6.1.3

运用分配律。

解题步骤 6.2

化简并合并同类项。

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解题步骤 6.2.1

化简每一项。

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解题步骤 6.2.1.1

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

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解题步骤 6.2.1.1.1

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

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

解题步骤 6.2.1.1.2

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 6.2.1.1.3

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 6.2.1.1.4

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 6.2.1.1.5

将 $1$ 和 $1$ 相加。

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

解题步骤 6.2.1.2

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

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解题步骤 6.2.1.2.1

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

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

解题步骤 6.2.1.2.2

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 6.2.1.2.3

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 6.2.1.2.4

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 6.2.1.2.5

将 $1$ 和 $1$ 相加。

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

解题步骤 6.2.1.3

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

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解题步骤 6.2.1.3.1

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

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

解题步骤 6.2.1.3.2

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 6.2.1.3.3

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 6.2.1.3.4

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 6.2.1.3.5

将 $1$ 和 $1$ 相加。

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

解题步骤 6.2.1.4

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

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解题步骤 6.2.1.4.1

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

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

解题步骤 6.2.1.4.2

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 6.2.1.4.3

对 $\sin (x)$ 进行 $1$ 次方运算。

解题步骤 6.2.1.4.4

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 6.2.1.4.5

将 $1$ 和 $1$ 相加。

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

解题步骤 6.2.1.4.6

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 6.2.1.4.7

对 $\cos (x)$ 进行 $1$ 次方运算。

解题步骤 6.2.1.4.8

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 6.2.1.4.9

将 $1$ 和 $1$ 相加。

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

解题步骤 6.2.2

在公分母上合并分子。

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

解题步骤 6.3

在公分母上合并分子。

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

解题步骤 6.4

将 $\sin (x)$ 和 $\sin (x)$ 相加。

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

解题步骤 6.5

使用完全平方法则进行因式分解。

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

解题步骤 7

将勾股恒等式反过来使用。

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

解题步骤 8

化简。

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解题步骤 8.1

将分子乘以分母的倒数。

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

解题步骤 8.2

化简分子。

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解题步骤 8.2.1

将 $1$ 重写为 $1^{2}$ 。

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

解题步骤 8.2.2

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 $b = \sin (x)$ 。

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

解题步骤 8.3

约去公因数。

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解题步骤 8.3.1

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

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

解题步骤 8.3.2

约去公因数。

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

解题步骤 8.3.3

重写表达式。

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

解题步骤 8.4

运用分配律。

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

解题步骤 8.5

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

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

解题步骤 8.6

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

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

解题步骤 8.7

在公分母上合并分子。

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

解题步骤 8.8

化简每一项。

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解题步骤 8.8.1

运用分配律。

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

解题步骤 8.8.2

将 $\sin (x)$ 乘以 $1$ 。

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

解题步骤 8.8.3

乘以 $\sin (x) (- \sin (x))$ 。

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解题步骤 8.8.3.1

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 8.8.3.2

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 8.8.3.3

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 8.8.3.4

将 $1$ 和 $1$ 相加。

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

解题步骤 8.9

将 $- \sin (x)$ 和 $\sin (x)$ 相加。

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

解题步骤 8.10

将 $1$ 和 $0$ 相加。

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

解题步骤 8.11

化简分子。

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解题步骤 8.11.1

将 $1$ 重写为 $1^{2}$ 。

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

解题步骤 8.11.2

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 $b = \sin (x)$ 。

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

解题步骤 8.12

约去 $1 + \sin (x)$ 的公因数。

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

解题步骤 9

因为两边已证明为相等，所以方程为恒等式。

$\frac{1}{\sec (x) + \tan (x)} = \frac{1 - \sin (x)}{\cos (x)}$ 是一个恒等式

初级微积分 示例

初级微积分示例