三角学示例

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

解题步骤 1

从等式两边同时减去 ${(\sec (x) - \tan (x))}^{2}$ 。

$\frac{1 - \sin (x)}{1 + \sin (x)} - {(\sec (x) - \tan (x))}^{2} = 0$

解题步骤 2

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

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解题步骤 2.1

化简每一项。

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解题步骤 2.1.1

化简每一项。

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解题步骤 2.1.1.1

将 $\sec (x)$ 重写为正弦和余弦形式。

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

解题步骤 2.1.1.2

将 $\tan (x)$ 重写为正弦和余弦形式。

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

解题步骤 2.1.2

将 ${(\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)})}^{2}$ 重写为 $(\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)}) (\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)})$ 。

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

解题步骤 2.1.3

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 2.1.3.1

运用分配律。

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

解题步骤 2.1.3.2

运用分配律。

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} (- \frac{\sin (x)}{\cos (x)}) - \frac{\sin (x)}{\cos (x)} (\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)})) = 0$

解题步骤 2.1.3.3

运用分配律。

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} (- \frac{\sin (x)}{\cos (x)}) - \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

解题步骤 2.1.4

化简并合并同类项。

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解题步骤 2.1.4.1

化简每一项。

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解题步骤 2.1.4.1.1

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

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解题步骤 2.1.4.1.1.1

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

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos (x) \cos (x)} + \frac{1}{\cos (x)} (- \frac{\sin (x)}{\cos (x)}) - \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

解题步骤 2.1.4.1.1.2

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 2.1.4.1.1.3

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 2.1.4.1.1.4

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 2.1.4.1.1.5

将 $1$ 和 $1$ 相加。

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

解题步骤 2.1.4.1.2

使用乘法的交换性质重写。

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

解题步骤 2.1.4.1.3

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

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解题步骤 2.1.4.1.3.1

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

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

解题步骤 2.1.4.1.3.2

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 2.1.4.1.3.3

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 2.1.4.1.3.4

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 2.1.4.1.3.5

将 $1$ 和 $1$ 相加。

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

解题步骤 2.1.4.1.4

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

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解题步骤 2.1.4.1.4.1

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

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

解题步骤 2.1.4.1.4.2

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 2.1.4.1.4.3

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 2.1.4.1.4.4

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 2.1.4.1.4.5

将 $1$ 和 $1$ 相加。

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

解题步骤 2.1.4.1.5

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

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解题步骤 2.1.4.1.5.1

将 $- 1$ 乘以 $- 1$ 。

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

解题步骤 2.1.4.1.5.2

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

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

解题步骤 2.1.4.1.5.3

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

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

解题步骤 2.1.4.1.5.4

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 2.1.4.1.5.5

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 2.1.4.1.5.6

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 2.1.4.1.5.7

将 $1$ 和 $1$ 相加。

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

解题步骤 2.1.4.1.5.8

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 2.1.4.1.5.9

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 2.1.4.1.5.10

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 2.1.4.1.5.11

将 $1$ 和 $1$ 相加。

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

解题步骤 2.1.4.2

从 $- \frac{\sin (x)}{\cos^{2} (x)}$ 中减去 $\frac{\sin (x)}{\cos^{2} (x)}$ 。

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

解题步骤 2.1.5

化简每一项。

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解题步骤 2.1.5.1

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

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

解题步骤 2.1.5.2

将负号移到分数的前面。

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

解题步骤 2.1.6

运用分配律。

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

解题步骤 2.1.7

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

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解题步骤 2.1.7.1

将 $- 1$ 乘以 $- 1$ 。

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

解题步骤 2.1.7.2

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

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

解题步骤 2.2

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

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

解题步骤 2.3

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

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

解题步骤 2.4

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 $(1 + \sin (x)) \cos^{2} (x)$ 的形式。

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解题步骤 2.4.1

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

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

解题步骤 2.4.2

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

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

解题步骤 2.4.3

重新排序 $(1 + \sin (x)) \cos^{2} (x)$ 的因式。

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

解题步骤 2.5

在公分母上合并分子。

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

解题步骤 2.6

化简每一项。

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解题步骤 2.6.1

化简分子。

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解题步骤 2.6.1.1

运用分配律。

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

解题步骤 2.6.1.2

将 $\cos^{2} (x)$ 乘以 $1$ 。

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

解题步骤 2.6.1.3

运用分配律。

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

解题步骤 2.6.1.4

将 $- 1$ 乘以 $1$ 。

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

解题步骤 2.6.1.5

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

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解题步骤 2.6.1.5.1

重新组合项。

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

解题步骤 2.6.1.5.2

添加圆括号。

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

解题步骤 2.6.1.5.3

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

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

解题步骤 2.6.1.5.4

从 $- \sin^{2} (x) - u$ 中分解出因数 $- 1$ 。

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解题步骤 2.6.1.5.4.1

将 $- \sin^{2} (x)$ 和 $- u$ 重新排序。

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

解题步骤 2.6.1.5.4.2

从 $- u$ 中分解出因数 $- 1$ 。

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

解题步骤 2.6.1.5.4.3

从 $- \sin^{2} (x)$ 中分解出因数 $- 1$ 。

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

解题步骤 2.6.1.5.4.4

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

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

解题步骤 2.6.1.5.5

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

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

解题步骤 2.6.1.5.6

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

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解题步骤 2.6.1.5.6.1

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

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

解题步骤 2.6.1.5.6.2

从 $\sin^{2} (x)$ 中分解出因数 $\sin (x)$ 。

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

解题步骤 2.6.1.5.6.3

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

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

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

解题步骤 2.6.1.5.7

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

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解题步骤 2.6.1.5.7.1

从 $- \sin (x)$ 中分解出因数 $- \sin (x)$ 。

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

解题步骤 2.6.1.5.7.2

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

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

解题步骤 2.6.2

将负号移到分数的前面。

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

解题步骤 2.7

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

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

解题步骤 2.8

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

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

解题步骤 2.9

在公分母上合并分子。

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

解题步骤 2.10

化简每一项。

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解题步骤 2.10.1

化简分子。

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解题步骤 2.10.1.1

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

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解题步骤 2.10.1.1.1

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

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

解题步骤 2.10.1.1.2

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

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

解题步骤 2.10.1.1.3

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

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

解题步骤 2.10.1.2

运用分配律。

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

解题步骤 2.10.1.3

化简。

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解题步骤 2.10.1.3.1

将 $- 1$ 乘以 $1$ 。

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

解题步骤 2.10.1.3.2

将 $- 1 \cos^{2} (x)$ 重写为 $- \cos^{2} (x)$ 。

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

解题步骤 2.10.1.3.3

将 $- 1 \sin (x)$ 重写为 $- \sin (x)$ 。

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

解题步骤 2.10.1.4

运用分配律。

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

解题步骤 2.10.1.5

将 $2$ 乘以 $1$ 。

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

解题步骤 2.10.1.6

将 $- 1$ 和 $2$ 相加。

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

解题步骤 2.10.1.7

将 $- \sin (x)$ 和 $2 \sin (x)$ 相加。

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

解题步骤 2.10.1.8

使用勾股恒等式。

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

解题步骤 2.10.1.9

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

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解题步骤 2.10.1.9.1

从 $\sin^{2} (x)$ 中分解出因数 $\sin (x)$ 。

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

解题步骤 2.10.1.9.2

乘以 $1$ 。

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

解题步骤 2.10.1.9.3

从 $\sin (x) \sin (x) + \sin (x) \cdot 1$ 中分解出因数 $\sin (x)$ 。

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

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

解题步骤 2.10.1.10

合并指数。

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解题步骤 2.10.1.10.1

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 2.10.1.10.2

对 $\sin (x)$ 进行 $1$ 次方运算。

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

解题步骤 2.10.1.10.3

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 2.10.1.10.4

将 $1$ 和 $1$ 相加。

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

解题步骤 2.10.2

约去 $\sin (x) + 1$ 和 $1 + \sin (x)$ 的公因数。

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解题步骤 2.10.2.1

重新排序项。

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

解题步骤 2.10.2.2

约去公因数。

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

解题步骤 2.10.2.3

重写表达式。

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

解题步骤 2.11

从 $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ 中减去 $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ 。

$0 = 0$

解题步骤 3

表达式的定义域是除使表达式无定义的值外的所有实数。在本例中，不存在使表达式无定义的实数。

三角学 示例

三角学示例