三角学示例

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

解题步骤 1

从左边开始。

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

解题步骤 2

化简。

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解题步骤 2.1

将 ${(\tan (x) + \cot (x))}^{2}$ 重写为 $(\tan (x) + \cot (x)) (\tan (x) + \cot (x))$ 。

$(\tan (x) + \cot (x)) (\tan (x) + \cot (x))$

解题步骤 2.2

使用 FOIL 方法展开 $(\tan (x) + \cot (x)) (\tan (x) + \cot (x))$ 。

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解题步骤 2.2.1

运用分配律。

$\tan (x) (\tan (x) + \cot (x)) + \cot (x) (\tan (x) + \cot (x))$

解题步骤 2.2.2

运用分配律。

$\tan (x) \tan (x) + \tan (x) \cot (x) + \cot (x) (\tan (x) + \cot (x))$

解题步骤 2.2.3

运用分配律。

$\tan (x) \tan (x) + \tan (x) \cot (x) + \cot (x) \tan (x) + \cot (x) \cot (x)$

解题步骤 2.3

化简并合并同类项。

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

解题步骤 3

将勾股恒等式反过来使用。

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

解题步骤 4

因数。

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解题步骤 4.1

将 $1$ 重写为 $1^{2}$ 。

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

解题步骤 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 $a = \sec (x)$ 和 $b = 1$ 。

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

解题步骤 5

将勾股恒等式反过来使用。

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

解题步骤 6

转换成正弦和余弦。

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解题步骤 6.1

对 $\sec (x)$ 使用倒数恒等式。

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

解题步骤 6.2

对 $\sec (x)$ 使用倒数恒等式。

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

解题步骤 6.3

使用商数恒等式以正弦和余弦书写 $\cot (x)$ 。

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

解题步骤 6.4

使用商数恒等式以正弦和余弦书写 $\tan (x)$ 。

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

解题步骤 6.5

对 $\csc (x)$ 使用倒数恒等式。

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

解题步骤 6.6

对 $\frac{1}{\sin (x)}$ 运用乘积法则。

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

解题步骤 7

化简。

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解题步骤 7.1

化简每一项。

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解题步骤 7.1.1

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

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解题步骤 7.1.1.1

运用分配律。

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

解题步骤 7.1.1.2

运用分配律。

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

解题步骤 7.1.1.3

运用分配律。

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

解题步骤 7.1.2

化简并合并同类项。

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解题步骤 7.1.2.1

化简每一项。

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解题步骤 7.1.2.1.1

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

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解题步骤 7.1.2.1.1.1

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

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

解题步骤 7.1.2.1.1.2

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 7.1.2.1.1.3

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 7.1.2.1.1.4

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 7.1.2.1.1.5

将 $1$ 和 $1$ 相加。

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

解题步骤 7.1.2.1.2

组合 $\frac{1}{\cos (x)}$ 和 $- 1$ 。

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

解题步骤 7.1.2.1.3

将负号移到分数的前面。

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

解题步骤 7.1.2.1.4

将 $\frac{1}{\cos (x)}$ 乘以 $1$ 。

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

解题步骤 7.1.2.1.5

将 $- 1$ 乘以 $1$ 。

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

解题步骤 7.1.2.2

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

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

解题步骤 7.1.2.3

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 ${\cos (x)}^{2}$ 的形式。

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解题步骤 7.1.2.3.1

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

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

解题步骤 7.1.2.3.2

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 7.1.2.3.3

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 7.1.2.3.4

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 7.1.2.3.5

将 $1$ 和 $1$ 相加。

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

解题步骤 7.1.2.4

在公分母上合并分子。

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

解题步骤 7.1.2.5

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

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

解题步骤 7.1.2.6

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 ${\cos (x)}^{2}$ 的形式。

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解题步骤 7.1.2.6.1

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

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

解题步骤 7.1.2.6.2

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 7.1.2.6.3

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 7.1.2.6.4

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 7.1.2.6.5

将 $1$ 和 $1$ 相加。

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

解题步骤 7.1.2.7

在公分母上合并分子。

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

解题步骤 7.1.2.8

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

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

解题步骤 7.1.2.9

组合 $- 1$ 和 $\frac{{\cos (x)}^{2}}{{\cos (x)}^{2}}$ 。

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

解题步骤 7.1.2.10

在公分母上合并分子。

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

解题步骤 7.1.3

化简分子。

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解题步骤 7.1.3.1

将 $- \cos (x)$ 和 $\cos (x)$ 相加。

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

解题步骤 7.1.3.2

将 $1$ 和 $0$ 相加。

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

解题步骤 7.1.3.3

将 $1$ 重写为 $1^{2}$ 。

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

解题步骤 7.1.3.4

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 $b = \cos (x)$ 。

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

解题步骤 7.1.4

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

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

解题步骤 7.1.5

约去 $\cos (x)$ 的公因数。

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解题步骤 7.1.5.1

从 $2 \cos (x)$ 中分解出因数 $\cos (x)$ 。

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

解题步骤 7.1.5.2

约去公因数。

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

解题步骤 7.1.5.3

重写表达式。

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

解题步骤 7.1.6

约去 $\sin (x)$ 的公因数。

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解题步骤 7.1.6.1

约去公因数。

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

解题步骤 7.1.6.2

重写表达式。

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

解题步骤 7.1.7

一的任意次幂都为一。

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

解题步骤 7.2

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

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

解题步骤 7.3

在公分母上合并分子。

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

解题步骤 7.4

化简分子。

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解题步骤 7.4.1

使用 FOIL 方法展开 $(1 + \cos (x)) (1 - \cos (x))$ 。

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解题步骤 7.4.1.1

运用分配律。

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

解题步骤 7.4.1.2

运用分配律。

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

解题步骤 7.4.1.3

运用分配律。

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

解题步骤 7.4.2

化简并合并同类项。

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解题步骤 7.4.2.1

化简每一项。

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解题步骤 7.4.2.1.1

将 $1$ 乘以 $1$ 。

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

解题步骤 7.4.2.1.2

将 $- \cos (x)$ 乘以 $1$ 。

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

解题步骤 7.4.2.1.3

将 $\cos (x)$ 乘以 $1$ 。

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

解题步骤 7.4.2.1.4

乘以 $\cos (x) (- \cos (x))$ 。

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解题步骤 7.4.2.1.4.1

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 7.4.2.1.4.2

对 $\cos (x)$ 进行 $1$ 次方运算。

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

解题步骤 7.4.2.1.4.3

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 7.4.2.1.4.4

将 $1$ 和 $1$ 相加。

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

解题步骤 7.4.2.2

将 $- \cos (x)$ 和 $\cos (x)$ 相加。

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

解题步骤 7.4.2.3

将 $1$ 和 $0$ 相加。

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

解题步骤 7.4.3

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

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

解题步骤 7.5

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

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

解题步骤 7.6

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

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

解题步骤 7.7

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 ${\cos (x)}^{2} {\sin (x)}^{2}$ 的形式。

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解题步骤 7.7.1

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

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

解题步骤 7.7.2

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

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

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

解题步骤 7.8

在公分母上合并分子。

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

解题步骤 7.9

化简分子。

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解题步骤 7.9.1

运用分配律。

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

解题步骤 7.9.2

将 ${\sin (x)}^{2}$ 乘以 $1$ 。

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

解题步骤 7.10

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

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

解题步骤 7.11

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

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

解题步骤 7.12

在公分母上合并分子。

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

解题步骤 7.13

化简分子。

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

解题步骤 8

现在，考虑等式的右边。

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

解题步骤 9

转换成正弦和余弦。

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解题步骤 9.1

对 $\sec (x)$ 使用倒数恒等式。

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

解题步骤 9.2

对 $\csc (x)$ 使用倒数恒等式。

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

解题步骤 9.3

对 $\frac{1}{\cos (x)}$ 运用乘积法则。

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

解题步骤 9.4

对 $\frac{1}{\sin (x)}$ 运用乘积法则。

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

解题步骤 10

化简每一项。

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

解题步骤 11

加上分数。

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解题步骤 11.1

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

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

解题步骤 11.2

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

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

解题步骤 11.3

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 11.3.1

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

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

解题步骤 11.3.2

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

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

解题步骤 11.3.3

重新排序 $\sin^{2} (x) \cos^{2} (x)$ 的因式。

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

解题步骤 11.4

在公分母上合并分子。

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

解题步骤 12

因为两边已证明为相等，所以方程为恒等式。

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

三角学 示例

三角学示例