三角学示例

tan (x) (sin (x) + cot (x) \cdot cos (x))

$\tan (x) (\sin (x) + \cot (x) \cdot \cos (x))$

解题步骤 1

将

tan (x)

$\tan (x)$ 重写为正弦和余弦形式。

\frac{sin (x)}{cos (x)} (sin (x) + cot (x) \cdot cos (x))

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

解题步骤 2

化简每一项。

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解题步骤 2.1

将

cot (x)

$\cot (x)$ 重写为正弦和余弦形式。

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

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

解题步骤 2.2

乘以

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

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

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解题步骤 2.2.1

组合

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

$\frac{\cos (x)}{\sin (x)}$ 和

cos (x)

$\cos (x)$ 。

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

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

解题步骤 2.2.2

对

cos (x)

$\cos (x)$ 进行

1

$1$ 次方运算。

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

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

解题步骤 2.2.3

对

cos (x)

$\cos (x)$ 进行

1

$1$ 次方运算。

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

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

解题步骤 2.2.4

使用幂法则

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

$a^{m} a^{n} = a^{m + n}$ 合并指数。

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

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

解题步骤 2.2.5

将

1

$1$ 和

1

$1$ 相加。

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

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

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

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

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

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

解题步骤 3

运用分配律。

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

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

解题步骤 4

乘以

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

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

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解题步骤 4.1

组合

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

$\frac{\sin (x)}{\cos (x)}$ 和

sin (x)

$\sin (x)$ 。

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

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

解题步骤 4.2

对

sin (x)

$\sin (x)$ 进行

1

$1$ 次方运算。

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

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

解题步骤 4.3

对

sin (x)

$\sin (x)$ 进行

1

$1$ 次方运算。

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

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

解题步骤 4.4

使用幂法则

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

$a^{m} a^{n} = a^{m + n}$ 合并指数。

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

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

解题步骤 4.5

将

1

$1$ 和

1

$1$ 相加。

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

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

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

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

解题步骤 5

合并。

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

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

解题步骤 6

化简每一项。

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解题步骤 6.1

约去

sin (x)

$\sin (x)$ 的公因数。

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解题步骤 6.1.1

约去公因数。

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

解题步骤 6.1.2

重写表达式。

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

解题步骤 6.2

约去

$\cos^{2} (x)$ 和

$\cos (x)$ 的公因数。

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解题步骤 6.2.1

从

$\cos^{2} (x)$ 中分解出因数

$\cos (x)$ 。

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

解题步骤 6.2.2

约去公因数。

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解题步骤 6.2.2.1

乘以

$1$ 。

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

解题步骤 6.2.2.2

约去公因数。

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

解题步骤 6.2.2.3

重写表达式。

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

解题步骤 6.2.2.4

用

$\cos (x)$ 除以

$1$ 。

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

解题步骤 7

化简每一项。

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解题步骤 7.1

从

$\sin^{2} (x)$ 中分解出因数

$\sin (x)$ 。

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

解题步骤 7.2

分离分数。

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

解题步骤 7.3

将

$\frac{\sin (x)}{\cos (x)}$ 转换成

$\tan (x)$ 。

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

解题步骤 7.4

用

$\sin (x)$ 除以

$1$ 。

$\sin (x) \tan (x) + \cos (x)$

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三角学 示例

三角学示例