三角学示例

$\cot (x) \sec^{4} (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

解题步骤 1

从右边开始。

$\cot (x) + 2 \tan (x) + \tan^{3} (x)$

解题步骤 2

转换成正弦和余弦。

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解题步骤 2.1

使用商数恒等式以正弦和余弦书写 $\cot (x)$ 。

$\frac{\cos (x)}{\sin (x)} + 2 \tan (x) + \tan^{3} (x)$

解题步骤 2.2

使用商数恒等式以正弦和余弦书写 $\tan (x)$ 。

$\frac{\cos (x)}{\sin (x)} + 2 \frac{\sin (x)}{\cos (x)} + \tan^{3} (x)$

解题步骤 2.3

使用商数恒等式以正弦和余弦书写 $\tan (x)$ 。

$\frac{\cos (x)}{\sin (x)} + 2 \frac{\sin (x)}{\cos (x)} + {(\frac{\sin (x)}{\cos (x)})}^{3}$

解题步骤 2.4

对 $\frac{\sin (x)}{\cos (x)}$ 运用乘积法则。

$\frac{\cos (x)}{\sin (x)} + 2 \frac{\sin (x)}{\cos (x)} + \frac{\sin^{3} (x)}{\cos^{3} (x)}$

解题步骤 3

化简。

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解题步骤 3.1

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

$\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x)}{\cos (x)} + \frac{{\sin (x)}^{3}}{{\cos (x)}^{3}}$

解题步骤 3.2

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

$\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x)}{\cos (x)} \cdot \frac{{\cos (x)}^{2}}{{\cos (x)}^{2}} + \frac{{\sin (x)}^{3}}{{\cos (x)}^{3}}$

解题步骤 3.3

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 ${\cos (x)}^{3}$ 的形式。

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解题步骤 3.3.1

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

$\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x) {\cos (x)}^{2}}{\cos (x) {\cos (x)}^{2}} + \frac{{\sin (x)}^{3}}{{\cos (x)}^{3}}$

解题步骤 3.3.2

通过指数相加将 $\cos (x)$ 乘以 ${\cos (x)}^{2}$ 。

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解题步骤 3.3.2.1

将 $\cos (x)$ 乘以 ${\cos (x)}^{2}$ 。

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解题步骤 3.3.2.1.1

对 $\cos (x)$ 进行 $1$ 次方运算。

$\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x) {\cos (x)}^{2}}{{\cos (x)}^{1} {\cos (x)}^{2}} + \frac{{\sin (x)}^{3}}{{\cos (x)}^{3}}$

解题步骤 3.3.2.1.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x) {\cos (x)}^{2}}{{\cos (x)}^{1 + 2}} + \frac{{\sin (x)}^{3}}{{\cos (x)}^{3}}$

解题步骤 3.3.2.2

将 $1$ 和 $2$ 相加。

$\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x) {\cos (x)}^{2}}{{\cos (x)}^{3}} + \frac{{\sin (x)}^{3}}{{\cos (x)}^{3}}$

解题步骤 3.4

在公分母上合并分子。

$\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x) {\cos (x)}^{2} + {\sin (x)}^{3}}{{\cos (x)}^{3}}$

解题步骤 3.5

从 $2 \sin (x) {\cos (x)}^{2} + {\sin (x)}^{3}$ 中分解出因数 $\sin (x)$ 。

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解题步骤 3.5.1

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

$\frac{\cos (x)}{\sin (x)} + \frac{\sin (x) (2 {\cos (x)}^{2}) + {\sin (x)}^{3}}{{\cos (x)}^{3}}$

解题步骤 3.5.2

从 ${\sin (x)}^{3}$ 中分解出因数 $\sin (x)$ 。

$\frac{\cos (x)}{\sin (x)} + \frac{\sin (x) (2 {\cos (x)}^{2}) + \sin (x) {\sin (x)}^{2}}{{\cos (x)}^{3}}$

解题步骤 3.5.3

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

$\frac{\cos (x)}{\sin (x)} + \frac{\sin (x) (2 {\cos (x)}^{2} + {\sin (x)}^{2})}{{\cos (x)}^{3}}$

解题步骤 3.6

要将 $\frac{\cos (x)}{\sin (x)}$ 写成带有公分母的分数，请乘以 $\frac{{\cos (x)}^{3}}{{\cos (x)}^{3}}$ 。

$\frac{\cos (x)}{\sin (x)} \cdot \frac{{\cos (x)}^{3}}{{\cos (x)}^{3}} + \frac{\sin (x) (2 {\cos (x)}^{2} + {\sin (x)}^{2})}{{\cos (x)}^{3}}$

解题步骤 3.7

要将 $\frac{\sin (x) (2 {\cos (x)}^{2} + {\sin (x)}^{2})}{{\cos (x)}^{3}}$ 写成带有公分母的分数，请乘以 $\frac{\sin (x)}{\sin (x)}$ 。

$\frac{\cos (x)}{\sin (x)} \cdot \frac{{\cos (x)}^{3}}{{\cos (x)}^{3}} + \frac{\sin (x) (2 {\cos (x)}^{2} + {\sin (x)}^{2})}{{\cos (x)}^{3}} \cdot \frac{\sin (x)}{\sin (x)}$

解题步骤 3.8

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 $\sin (x) {\cos (x)}^{3}$ 的形式。

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解题步骤 3.8.1

将 $\frac{\cos (x)}{\sin (x)}$ 乘以 $\frac{{\cos (x)}^{3}}{{\cos (x)}^{3}}$ 。

$\frac{\cos (x) {\cos (x)}^{3}}{\sin (x) {\cos (x)}^{3}} + \frac{\sin (x) (2 {\cos (x)}^{2} + {\sin (x)}^{2})}{{\cos (x)}^{3}} \cdot \frac{\sin (x)}{\sin (x)}$

解题步骤 3.8.2

将 $\frac{\sin (x) (2 {\cos (x)}^{2} + {\sin (x)}^{2})}{{\cos (x)}^{3}}$ 乘以 $\frac{\sin (x)}{\sin (x)}$ 。

$\frac{\cos (x) {\cos (x)}^{3}}{\sin (x) {\cos (x)}^{3}} + \frac{\sin (x) (2 {\cos (x)}^{2} + {\sin (x)}^{2}) \sin (x)}{{\cos (x)}^{3} \sin (x)}$

解题步骤 3.8.3

重新排序 $\sin (x) {\cos (x)}^{3}$ 的因式。

$\frac{\cos (x) {\cos (x)}^{3}}{{\cos (x)}^{3} \sin (x)} + \frac{\sin (x) (2 {\cos (x)}^{2} + {\sin (x)}^{2}) \sin (x)}{{\cos (x)}^{3} \sin (x)}$

解题步骤 3.9

在公分母上合并分子。

$\frac{\cos (x) {\cos (x)}^{3} + \sin (x) (2 {\cos (x)}^{2} + {\sin (x)}^{2}) \sin (x)}{{\cos (x)}^{3} \sin (x)}$

解题步骤 3.10

化简分子。

$\frac{{(\cos^{2} (x) + \sin^{2} (x))}^{2}}{\cos^{3} (x) \sin (x)}$

解题步骤 4

使用勾股恒等式。

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解题步骤 4.1

重新整理项。

$\frac{{(\sin^{2} (x) + \cos^{2} (x))}^{2}}{\cos^{3} (x) \sin (x)}$

解题步骤 4.2

使用勾股恒等式。

$\frac{1^{2}}{\cos^{3} (x) \sin (x)}$

解题步骤 5

一的任意次幂都为一。

$\frac{1}{\cos^{3} (x) \sin (x)}$

解题步骤 6

将 $\frac{1}{\cos^{3} (x) \sin (x)}$ 重写为 $\cot (x) \sec^{4} (x)$ 。

$\cot (x) \sec^{4} (x)$

解题步骤 7

因为两边已证明为相等，所以方程为恒等式。

$\cot (x) \sec^{4} (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$ 是一个恒等式

三角学 示例

三角学示例