代数示例

\frac{1}{1 + \tan^{2} (x)} + \frac{1}{1 + \cot^{2} (x)}

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

解题步骤 1

重新整理项。

\frac{1}{\tan^{2} (x) + 1} + \frac{1}{1 + \cot^{2} (x)}

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

解题步骤 2

使用勾股恒等式。

\frac{1}{\sec^{2} (x)} + \frac{1}{1 + \cot^{2} (x)}

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

解题步骤 3

使用勾股恒等式。

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

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

解题步骤 4

化简每一项。

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解题步骤 4.1

将

1

$1$ 重写为

1^{2}

$1^{2}$ 。

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

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

解题步骤 4.2

将

\frac{1^{2}}{\sec^{2} (x)}

$\frac{1^{2}}{\sec^{2} (x)}$ 重写为

{(\frac{1}{\sec (x)})}^{2}

${(\frac{1}{\sec (x)})}^{2}$ 。

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

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

解题步骤 4.3

将

\sec (x)

$\sec (x)$ 重写为正弦和余弦形式。

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

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

解题步骤 4.4

乘以分数的倒数从而实现除以

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

$\frac{1}{\cos (x)}$ 。

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

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

解题步骤 4.5

将

\cos (x)

$\cos (x)$ 乘以

1

$1$ 。

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

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

解题步骤 4.6

将

1

$1$ 重写为

1^{2}

$1^{2}$ 。

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

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

解题步骤 4.7

将

\frac{1^{2}}{\csc^{2} (x)}

$\frac{1^{2}}{\csc^{2} (x)}$ 重写为

{(\frac{1}{\csc (x)})}^{2}

${(\frac{1}{\csc (x)})}^{2}$ 。

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

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

解题步骤 4.8

将

\csc (x)

$\csc (x)$ 重写为正弦和余弦形式。

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

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

解题步骤 4.9

乘以分数的倒数从而实现除以

\frac{1}{\sin (x)}

$\frac{1}{\sin (x)}$ 。

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

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

解题步骤 4.10

将

\sin (x)

$\sin (x)$ 乘以

1

$1$ 。

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

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

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

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

解题步骤 5

重新整理项。

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

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

解题步骤 6

使用勾股恒等式。

1

$1$

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