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受欢迎的三角函数 >

arctan(1+x)+arctan(1-x)=arctan(1/2)

解答

arctan (1 + x) + arctan (1 - x) = arctan (\frac{1}{2})

解答

x = 2, x = - 2

求解步骤

arctan (1 + x) + arctan (1 - x) = arctan (\frac{1}{2})

使用三角恒等式改写

arctan (1 + x) + arctan (1 - x)

使用和差化积恒等式:

arctan (s) + arctan (t) = arctan (\frac{s + t}{1 - s t})

= arctan (\frac{1 + x + 1 - x}{1 - ( 1 + x ) ( 1 - x )})

arctan (\frac{1 + x + 1 - x}{1 - ( 1 + x ) ( 1 - x )}) = arctan (\frac{1}{2})

使用反三角函数性质

arctan (\frac{1 + x + 1 - x}{1 - ( 1 + x ) ( 1 - x )}) = arctan (\frac{1}{2})

arctan (x) = a \Rightarrow x = tan (a)

\frac{1 + x + 1 - x}{1 - ( 1 + x ) ( 1 - x )} = tan (arctan (\frac{1}{2}))

tan (arctan (\frac{1}{2})) = \frac{1}{2}

tan (arctan (\frac{1}{2}))

使用三角恒等式改写:

tan (arctan (\frac{1}{2})) = \frac{1}{2}

利用以下特性:

tan (arctan (x)) = x

= \frac{1}{2}

= \frac{1}{2}

\frac{1 + x + 1 - x}{1 - ( 1 + x ) ( 1 - x )} = \frac{1}{2}

\frac{1 + x + 1 - x}{1 - ( 1 + x ) ( 1 - x )} = \frac{1}{2}

解

\frac{1 + x + 1 - x}{1 - ( 1 + x ) ( 1 - x )} = \frac{1}{2} : x = 2, x = - 2

\frac{1 + x + 1 - x}{1 - ( 1 + x ) ( 1 - x )} = \frac{1}{2}

使用分式交叉相乘

\frac{1 + x + 1 - x}{1 - ( 1 + x ) ( 1 - x )} = \frac{1}{2}

化简

\frac{1 + x + 1 - x}{1 - ( 1 + x ) ( 1 - x )} : \frac{2}{x ^{2}}

\frac{1 + x + 1 - x}{1 - ( 1 + x ) ( 1 - x )}

1 + x + 1 - x = 2

1 + x + 1 - x

对同类项分组

= x - x + 1 + 1

同类项相加:

x - x = 0

= 1 + 1

数字相加:

1 + 1 = 2

= 2

= \frac{2}{1 - ( x + 1 ) ( - x + 1 )}

乘开

1 - (1 + x) (1 - x) : x^{2}

1 - (1 + x) (1 - x)

乘开

- (1 + x) (1 - x) : - 1 + x^{2}

乘开

(1 + x) (1 - x) : 1 - x^{2}

(1 + x) (1 - x)

使用平方差公式:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = x

= 1^{2} - x^{2}

使用法则

1^{a} = 1

1^{2} = 1

= 1 - x^{2}

= - (1 - x^{2})

打开括号

= - (1) - (- x^{2})

使用加减运算法则

- (- a) = a, - (a) = - a

= - 1 + x^{2}

= 1 - 1 + x^{2}

1 - 1 = 0

= x^{2}

= \frac{2}{x ^{2}}

\frac{2}{x ^{2}} = \frac{1}{2}

使用分式交叉相乘: 若

\frac{a}{b} = \frac{c}{d}

则

a \cdot d = b \cdot c

2 \cdot 2 = x^{2} \cdot 1

化简

2 \cdot 2 = x^{2} \cdot 1

化简

2 \cdot 2 : 4

2 \cdot 2

数字相乘:

2 \cdot 2 = 4

= 4

化简

x^{2} \cdot 1 : x^{2}

x^{2} \cdot 1

乘以:

x^{2} \cdot 1 = x^{2}

= x^{2}

4 = x^{2}

4 = x^{2}

4 = x^{2}

解

4 = x^{2} : x = 2, x = - 2

4 = x^{2}

交换两边

x^{2} = 4

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

x = 4, x = - 4

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

a^{2} = a, a \geq 0

2^{2} = 2

= 2

- 4 = - 2

- 4

因式分解数字:

4 = 2^{2}

= - 2^{2}

使用根式运算法则:

a^{2} = a, a \geq 0

2^{2} = - 2

= - 2

x = 2, x = - 2

x = 2, x = - 2

验证解

找到无定义的点（奇点）:

x = 0

取

\frac{1 + x + 1 - x}{1 - ( 1 + x ) ( 1 - x )}

的分母，令其等于零

解

1 - (1 + x) (1 - x) = 0 : x = 0

1 - (1 + x) (1 - x) = 0

展开

1 - (1 + x) (1 - x) : x^{2}

1 - (1 + x) (1 - x)

乘开

- (1 + x) (1 - x) : - 1 + x^{2}

乘开

(1 + x) (1 - x) : 1 - x^{2}

(1 + x) (1 - x)

使用平方差公式:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = x

= 1^{2} - x^{2}

使用法则

1^{a} = 1

1^{2} = 1

= 1 - x^{2}

= - (1 - x^{2})

打开括号

= - (1) - (- x^{2})

使用加减运算法则

- (- a) = a, - (a) = - a

= - 1 + x^{2}

= 1 - 1 + x^{2}

1 - 1 = 0

= x^{2}

x^{2} = 0

使用求根公式求解

x^{2} = 0

二次方程求根公式:

若

a = 1, b = 0, c = 0

x_{1, 2} = \frac{- 0 \pm 0 ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

x_{1, 2} = \frac{- 0 \pm 0 ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

0^{2} - 4 \cdot 1 \cdot 0 = 0

0^{2} - 4 \cdot 1 \cdot 0

使用法则

0^{a} = 0

0^{2} = 0

= 0 - 4 \cdot 1 \cdot 0

使用法则

0 \cdot a = 0

= 0 - 0

数字相减:

0 - 0 = 0

= 0

x_{1, 2} = \frac{- 0 \pm 0}{2 \cdot 1}

x = \frac{- 0}{2 \cdot 1}

\frac{- 0}{2 \cdot 1} = 0

\frac{- 0}{2 \cdot 1}

= \frac{0}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{0}{2}

使用法则

\frac{0}{a} = 0, a \neq = 0

= 0

x = 0

二次方程组的解是:

x = 0

以下点无定义

x = 0

将不在定义域的点与解相综合:

x = 2, x = - 2

x = 2, x = - 2

将解代入原方程进行验证

将它们代入

arctan (1 + x) + arctan (1 - x) = arctan (\frac{1}{2})

检验解是否符合

去除与方程不符的解。

检验

2

的解:

真

2

代入

n = 1

2

对于

arctan (1 + x) + arctan (1 - x) = arctan (\frac{1}{2})

代入

x = 2

arctan (1 + 2) + arctan (1 - 2) = arctan (\frac{1}{2})

整理后得

0.46364 \dots = 0.46364 \dots

\Rightarrow 真

检验

- 2

的解:

真

- 2

代入

n = 1

- 2

对于

arctan (1 + x) + arctan (1 - x) = arctan (\frac{1}{2})

代入

x = - 2

arctan (1 - 2) + arctan (1 - (- 2)) = arctan (\frac{1}{2})

整理后得

0.46364 \dots = 0.46364 \dots

\Rightarrow 真

x = 2, x = - 2

作图

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