三角学示例

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

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

解题步骤 1

化简每一项。

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解题步骤 1.1

将

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

$\frac{\tan^{2} (x)}{\sec^{2} (x)}$ 重写为

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

${(\frac{\tan (x)}{\sec (x)})}^{2}$ 。

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

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

解题步骤 1.2

将

sec (x)

$\sec (x)$ 重写为正弦和余弦形式。

{⎛ ⎝ \frac{tan (x)}{\frac{1}{cos (x)}} ⎞ ⎠}^{2} + \frac{{cot}^{2} (x)}{{csc}^{2} (x)}

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

解题步骤 1.3

将

tan (x)

$\tan (x)$ 重写为正弦和余弦形式。

{⎛ ⎜ ⎝ \frac{\frac{sin (x)}{cos (x)}}{\frac{1}{cos (x)}} ⎞ ⎟ ⎠}^{2} + \frac{{cot}^{2} (x)}{{csc}^{2} (x)}

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

解题步骤 1.4

乘以分数的倒数从而实现除以

\frac{1}{cos (x)}

$\frac{1}{\cos (x)}$ 。

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

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

解题步骤 1.5

将

cos (x)

$\cos (x)$ 写成分母为

1

$1$ 的分数。

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

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

解题步骤 1.6

约去

cos (x)

$\cos (x)$ 的公因数。

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解题步骤 1.6.1

约去公因数。

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

解题步骤 1.6.2

重写表达式。

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

解题步骤 1.7

将

$\frac{\cot^{2} (x)}{\csc^{2} (x)}$ 重写为

${(\frac{\cot (x)}{\csc (x)})}^{2}$ 。

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

解题步骤 1.8

将

$\csc (x)$ 重写为正弦和余弦形式。

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

解题步骤 1.9

将

$\cot (x)$ 重写为正弦和余弦形式。

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

解题步骤 1.10

乘以分数的倒数从而实现除以

$\frac{1}{\sin (x)}$ 。

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

解题步骤 1.11

将

$\sin (x)$ 写成分母为

$1$ 的分数。

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

解题步骤 1.12

约去

$\sin (x)$ 的公因数。

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解题步骤 1.12.1

约去公因数。

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

解题步骤 1.12.2

重写表达式。

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

解题步骤 2

使用勾股恒等式。

$1$

三角学 示例

三角学示例