初级微积分示例

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

解题步骤 1

化简分子。

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解题步骤 1.1

将 $\tan (x)$ 重写为正弦和余弦形式。

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

解题步骤 1.2

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

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

解题步骤 1.3

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

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解题步骤 1.3.1

运用分配律。

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

解题步骤 1.3.2

运用分配律。

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

解题步骤 1.3.3

运用分配律。

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

解题步骤 1.4

化简并合并同类项。

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解题步骤 1.4.1

化简每一项。

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解题步骤 1.4.1.1

乘以 $\sin (x) \sin (x)$ 。

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解题步骤 1.4.1.1.1

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 1.4.1.1.2

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 1.4.1.1.3

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 1.4.1.1.4

将 $1$ 和 $1$ 相加。

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

解题步骤 1.4.1.2

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

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解题步骤 1.4.1.2.1

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

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

解题步骤 1.4.1.2.2

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 1.4.1.2.3

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 1.4.1.2.4

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 1.4.1.2.5

将 $1$ 和 $1$ 相加。

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

解题步骤 1.4.1.3

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

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解题步骤 1.4.1.3.1

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

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

解题步骤 1.4.1.3.2

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 1.4.1.3.3

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 1.4.1.3.4

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 1.4.1.3.5

将 $1$ 和 $1$ 相加。

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

解题步骤 1.4.1.4

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

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解题步骤 1.4.1.4.1

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

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

解题步骤 1.4.1.4.2

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 1.4.1.4.3

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 1.4.1.4.4

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 1.4.1.4.5

将 $1$ 和 $1$ 相加。

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

解题步骤 1.4.1.4.6

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 1.4.1.4.7

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 1.4.1.4.8

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 1.4.1.4.9

将 $1$ 和 $1$ 相加。

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

解题步骤 1.4.2

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

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

解题步骤 1.5

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

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

解题步骤 1.6

将 $\sec (x)$ 重写为正弦和余弦形式。

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

解题步骤 1.7

对 $\frac{1}{\cos (x)}$ 运用乘积法则。

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

解题步骤 1.8

一的任意次幂都为一。

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

解题步骤 1.9

以因式分解的形式重写 $\sin^{2} (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)} + \cos^{2} (x) - \frac{1}{\cos^{2} (x)}$ 。

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解题步骤 1.9.1

使 $u = \frac{2 \sin^{2} (x)}{\cos (x)}$ 。用 $u$ 代入替换所有出现的 $\frac{2 \sin^{2} (x)}{\cos (x)}$ 。

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解题步骤 1.9.1.1

从 $1$ 中减去 $1$ 。

$\frac{u + 0}{\tan (x)}$

解题步骤 1.9.1.2

将 $u$ 和 $0$ 相加。

$\frac{u}{\tan (x)}$

解题步骤 1.9.2

使用 $\frac{2 \sin^{2} (x)}{\cos (x)}$ 替换所有出现的 $u$ 。

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

解题步骤 2

将 $\tan (x)$ 重写为正弦和余弦形式。

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

解题步骤 3

将分子乘以分母的倒数。

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

解题步骤 4

合并。

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

解题步骤 5

约去 $\sin^{2} (x)$ 和 $\sin (x)$ 的公因数。

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解题步骤 5.1

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

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

解题步骤 5.2

约去公因数。

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解题步骤 5.2.1

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

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

解题步骤 5.2.2

约去公因数。

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

解题步骤 5.2.3

重写表达式。

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

解题步骤 6

约去 $\cos (x)$ 的公因数。

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解题步骤 6.1

约去公因数。

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

解题步骤 6.2

用 $2 \sin (x)$ 除以 $1$ 。

$2 \sin (x)$

初级微积分 示例

初级微积分示例