三角学示例

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

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

解题步骤 1

化简每一项。

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解题步骤 1.1

将

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

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

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

${(\frac{\tan (t)}{\sec (t)})}^{2}$ 。

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

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

解题步骤 1.2

将

sec (t)

$\sec (t)$ 重写为正弦和余弦形式。

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

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

解题步骤 1.3

将

tan (t)

$\tan (t)$ 重写为正弦和余弦形式。

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

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

解题步骤 1.4

乘以分数的倒数从而实现除以

\frac{1}{cos (t)}

$\frac{1}{\cos (t)}$ 。

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

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

解题步骤 1.5

将

cos (t)

$\cos (t)$ 写成分母为

1

$1$ 的分数。

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

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

解题步骤 1.6

约去

cos (t)

$\cos (t)$ 的公因数。

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解题步骤 1.6.1

约去公因数。

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

解题步骤 1.6.2

重写表达式。

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

解题步骤 1.7

将

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

${(\frac{\cot (t)}{\csc (t)})}^{2}$ 。

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

解题步骤 1.8

将

$\csc (t)$ 重写为正弦和余弦形式。

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

解题步骤 1.9

将

$\cot (t)$ 重写为正弦和余弦形式。

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

解题步骤 1.10

乘以分数的倒数从而实现除以

$\frac{1}{\sin (t)}$ 。

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

解题步骤 1.11

将

$\sin (t)$ 写成分母为

$1$ 的分数。

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

解题步骤 1.12

约去

$\sin (t)$ 的公因数。

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解题步骤 1.12.1

约去公因数。

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

解题步骤 1.12.2

重写表达式。

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

解题步骤 2

使用勾股恒等式。

$1$

三角学 示例

三角学示例