三角学示例

(sec (x) + tan (x)) (1 - sin (x))

$(\sec (x) + \tan (x)) (1 - \sin (x))$

解题步骤 1

化简每一项。

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解题步骤 1.1

将

sec (x)

$\sec (x)$ 重写为正弦和余弦形式。

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

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

解题步骤 1.2

将

tan (x)

$\tan (x)$ 重写为正弦和余弦形式。

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

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

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

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

解题步骤 2

使用 FOIL 方法展开

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

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

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解题步骤 2.1

运用分配律。

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

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

解题步骤 2.2

运用分配律。

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

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

解题步骤 2.3

运用分配律。

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

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

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

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

解题步骤 3

化简并合并同类项。

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解题步骤 3.1

化简每一项。

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解题步骤 3.1.1

将

\frac{1}{cos (x)}

$\frac{1}{\cos (x)}$ 乘以

1

$1$ 。

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

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

解题步骤 3.1.2

使用乘法的交换性质重写。

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

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

解题步骤 3.1.3

组合

sin (x)

$\sin (x)$ 和

\frac{1}{cos (x)}

$\frac{1}{\cos (x)}$ 。

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

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

解题步骤 3.1.4

将

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

$\frac{\sin (x)}{\cos (x)}$ 乘以

1

$1$ 。

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

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

解题步骤 3.1.5

使用乘法的交换性质重写。

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

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

解题步骤 3.1.6

乘以

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

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

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解题步骤 3.1.6.1

组合

sin (x)

$\sin (x)$ 和

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

$\frac{\sin (x)}{\cos (x)}$ 。

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

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

解题步骤 3.1.6.2

对

sin (x)

$\sin (x)$ 进行

1

$1$ 次方运算。

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

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

解题步骤 3.1.6.3

对

sin (x)

$\sin (x)$ 进行

1

$1$ 次方运算。

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

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

解题步骤 3.1.6.4

使用幂法则

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ 合并指数。

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

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

解题步骤 3.1.6.5

将

1

$1$ 和

1

$1$ 相加。

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

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

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

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

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

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

解题步骤 3.2

将

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

$- \frac{\sin (x)}{\cos (x)}$ 和

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

$\frac{\sin (x)}{\cos (x)}$ 相加。

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

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

解题步骤 3.3

将

\frac{1}{cos (x)}

$\frac{1}{\cos (x)}$ 和

0

$0$ 相加。

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

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

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

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

解题步骤 4

在公分母上合并分子。

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

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

解题步骤 5

使用勾股恒等式。

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

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

解题步骤 6

约去

{cos}^{2} (x)

$\cos^{2} (x)$ 和

cos (x)

$\cos (x)$ 的公因数。

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解题步骤 6.1

从

{cos}^{2} (x)

$\cos^{2} (x)$ 中分解出因数

cos (x)

$\cos (x)$ 。

\frac{cos (x) cos (x)}{cos (x)}

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

解题步骤 6.2

约去公因数。

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解题步骤 6.2.1

乘以

1

$1$ 。

\frac{cos (x) cos (x)}{cos (x) \cdot 1}

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

解题步骤 6.2.2

约去公因数。

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

解题步骤 6.2.3

重写表达式。

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

解题步骤 6.2.4

用

$\cos (x)$ 除以

$1$ 。

$\cos (x)$

三角学 示例

三角学示例