三角学示例

\frac{2 sin (t) cos (t)}{sin (t) + cos (t)} = sin (t) + cos (t) - \frac{1}{sin (t) + cos (t)}

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

解题步骤 1

从右边开始。

sin (t) + cos (t) - \frac{1}{sin (t) + cos (t)}

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

解题步骤 2

化简表达式。

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解题步骤 2.1

要将

sin (t)

$\sin (t)$ 写成带有公分母的分数，请乘以

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

$\frac{\sin (t) + \cos (t)}{\sin (t) + \cos (t)}$ 。

\frac{sin (t) (sin (t) + cos (t))}{sin (t) + cos (t)} - \frac{1}{sin (t) + cos (t)} + cos (t)

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

解题步骤 2.2

在公分母上合并分子。

\frac{sin (t) (sin (t) + cos (t)) - 1}{sin (t) + cos (t)} + cos (t)

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

解题步骤 2.3

化简分子。

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解题步骤 2.3.1

运用分配律。

\frac{sin (t) sin (t) + sin (t) cos (t) - 1}{sin (t) + cos (t)} + cos (t)

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

解题步骤 2.3.2

乘以

sin (t) sin (t)

$\sin (t) \sin (t)$ 。

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解题步骤 2.3.2.1

对

sin (t)

$\sin (t)$ 进行

1

$1$ 次方运算。

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

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

解题步骤 2.3.2.2

对

sin (t)

$\sin (t)$ 进行

1

$1$ 次方运算。

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

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

解题步骤 2.3.2.3

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 (t)}^{1 + 1} + sin (t) cos (t) - 1}{sin (t) + cos (t)} + cos (t)

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

解题步骤 2.3.2.4

将

1

$1$ 和

1

$1$ 相加。

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

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

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

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

解题步骤 2.3.3

移动

- 1

$- 1$ 。

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

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

解题步骤 2.3.4

将

{sin}^{2} (t)

$\sin^{2} (t)$ 和

- 1

$- 1$ 重新排序。

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

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

解题步骤 2.3.5

将

- 1

$- 1$ 重写为

- 1 (1)

$- 1 (1)$ 。

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

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

解题步骤 2.3.6

从

{sin}^{2} (t)

$\sin^{2} (t)$ 中分解出因数

- 1

$- 1$ 。

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

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

解题步骤 2.3.7

从

- 1 (1) - 1 (- {sin}^{2} (t))

$- 1 (1) - 1 (- \sin^{2} (t))$ 中分解出因数

- 1

$- 1$ 。

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

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

解题步骤 2.3.8

将

- 1 (1 - {sin}^{2} (t))

$- 1 (1 - \sin^{2} (t))$ 重写为

- (1 - {sin}^{2} (t))

$- (1 - \sin^{2} (t))$ 。

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

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

解题步骤 2.3.9

使用勾股恒等式。

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

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

解题步骤 2.3.10

从

- {cos}^{2} (t) + sin (t) cos (t)

$- \cos^{2} (t) + \sin (t) \cos (t)$ 中分解出因数

cos (t)

$\cos (t)$ 。

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解题步骤 2.3.10.1

从

- {cos}^{2} (t)

$- \cos^{2} (t)$ 中分解出因数

cos (t)

$\cos (t)$ 。

\frac{cos (t) (- cos (t)) + sin (t) cos (t)}{sin (t) + cos (t)} + cos (t)

$\frac{\cos (t) (- \cos (t)) + \sin (t) \cos (t)}{\sin (t) + \cos (t)} + \cos (t)$

解题步骤 2.3.10.2

从

sin (t) cos (t)

$\sin (t) \cos (t)$ 中分解出因数

cos (t)

$\cos (t)$ 。

\frac{cos (t) (- cos (t)) + cos (t) sin (t)}{sin (t) + cos (t)} + cos (t)

$\frac{\cos (t) (- \cos (t)) + \cos (t) \sin (t)}{\sin (t) + \cos (t)} + \cos (t)$

解题步骤 2.3.10.3

从

cos (t) (- cos (t)) + cos (t) sin (t)

$\cos (t) (- \cos (t)) + \cos (t) \sin (t)$ 中分解出因数

cos (t)

$\cos (t)$ 。

\frac{cos (t) (- cos (t) + sin (t))}{sin (t) + cos (t)} + cos (t)

$\frac{\cos (t) (- \cos (t) + \sin (t))}{\sin (t) + \cos (t)} + \cos (t)$

\frac{cos (t) (- cos (t) + sin (t))}{sin (t) + cos (t)} + cos (t)

$\frac{\cos (t) (- \cos (t) + \sin (t))}{\sin (t) + \cos (t)} + \cos (t)$

\frac{cos (t) (- cos (t) + sin (t))}{sin (t) + cos (t)} + cos (t)

$\frac{\cos (t) (- \cos (t) + \sin (t))}{\sin (t) + \cos (t)} + \cos (t)$

解题步骤 2.4

要将

cos (t)

$\cos (t)$ 写成带有公分母的分数，请乘以

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

$\frac{\sin (t) + \cos (t)}{\sin (t) + \cos (t)}$ 。

\frac{cos (t) (- cos (t) + sin (t))}{sin (t) + cos (t)} + \frac{cos (t) (sin (t) + cos (t))}{sin (t) + cos (t)}

$\frac{\cos (t) (- \cos (t) + \sin (t))}{\sin (t) + \cos (t)} + \frac{\cos (t) (\sin (t) + \cos (t))}{\sin (t) + \cos (t)}$

解题步骤 2.5

在公分母上合并分子。

\frac{cos (t) (- cos (t) + sin (t)) + cos (t) (sin (t) + cos (t))}{sin (t) + cos (t)}

$\frac{\cos (t) (- \cos (t) + \sin (t)) + \cos (t) (\sin (t) + \cos (t))}{\sin (t) + \cos (t)}$

解题步骤 2.6

化简分子。

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解题步骤 2.6.1

从

cos (t) (- cos (t) + sin (t)) + cos (t) (sin (t) + cos (t))

$\cos (t) (- \cos (t) + \sin (t)) + \cos (t) (\sin (t) + \cos (t))$ 中分解出因数

cos (t)

$\cos (t)$ 。

\frac{cos (t) (- cos (t) + sin (t) + sin (t) + cos (t))}{sin (t) + cos (t)}

$\frac{\cos (t) (- \cos (t) + \sin (t) + \sin (t) + \cos (t))}{\sin (t) + \cos (t)}$

解题步骤 2.6.2

将

- cos (t)

$- \cos (t)$ 和

cos (t)

$\cos (t)$ 相加。

\frac{cos (t) (0 + sin (t) + sin (t))}{sin (t) + cos (t)}

$\frac{\cos (t) (0 + \sin (t) + \sin (t))}{\sin (t) + \cos (t)}$

解题步骤 2.6.3

将

0

$0$ 和

sin (t)

$\sin (t)$ 相加。

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

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

解题步骤 2.6.4

将

sin (t)

$\sin (t)$ 和

sin (t)

$\sin (t)$ 相加。

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

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

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

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

解题步骤 2.7

将

2

$2$ 移到

cos (t)

$\cos (t)$ 的左侧。

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

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

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

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

解题步骤 3

重新排序项。

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

解题步骤 4

将

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

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

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

解题步骤 5

因为两边已证明为相等，所以方程为恒等式。

$\frac{2 \sin (t) \cos (t)}{\sin (t) + \cos (t)} = \sin (t) + \cos (t) - \frac{1}{\sin (t) + \cos (t)}$ 是一个恒等式

三角学 示例

三角学示例