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受欢迎的三角函数 >

cot^2(x)=sec^2(x)-1

解答

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

解答

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

+1

度数

x = 4 5^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

求解步骤

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

两边减去

sec^{2} (x) - 1

cot^{2} (x) - sec^{2} (x) + 1 = 0

使用三角恒等式改写

1 + cot^{2} (x) - sec^{2} (x)

使用毕达哥拉斯恒等式:

sec^{2} (x) = tan^{2} (x) + 1

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

= cot^{2} (x) - tan^{2} (x)

cot^{2} (x) - tan^{2} (x) = 0

分解

cot^{2} (x) - tan^{2} (x) : (cot (x) + tan (x)) (cot (x) - tan (x))

cot^{2} (x) - tan^{2} (x)

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

cot^{2} (x) - tan^{2} (x) = (cot (x) + tan (x)) (cot (x) - tan (x))

= (cot (x) + tan (x)) (cot (x) - tan (x))

(cot (x) + tan (x)) (cot (x) - tan (x)) = 0

分别求解每个部分

cot (x) + tan (x) = 0 or cot (x) - tan (x) = 0

cot (x) + tan (x) = 0 :

无解

cot (x) + tan (x) = 0

使用三角恒等式改写

cot (x) + tan (x)

使用基本三角恒等式:

tan (x) = \frac{1}{cot ( x )}

= cot (x) + \frac{1}{cot ( x )}

cot (x) + \frac{1}{cot ( x )} = 0

用替代法求解

cot (x) + \frac{1}{cot ( x )} = 0

令

: cot (x) = u

u + \frac{1}{u} = 0

u + \frac{1}{u} = 0 : u = i, u = - i

u + \frac{1}{u} = 0

在两边乘以

u

u + \frac{1}{u} = 0

在两边乘以

u

uu + \frac{1}{u} u = 0 \cdot u

化简

uu + \frac{1}{u} u = 0 \cdot u

化简

uu : u^{2}

uu

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= u^{1 + 1}

数字相加:

1 + 1 = 2

= u^{2}

化简

\frac{1}{u} u : 1

\frac{1}{u} u

分式相乘:

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

= \frac{1 \cdot u}{u}

约分:

u

= 1

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

u^{2} + 1 = 0

u^{2} + 1 = 0

u^{2} + 1 = 0

解

u^{2} + 1 = 0 : u = i, u = - i

u^{2} + 1 = 0

将

1

到右边

u^{2} + 1 = 0

两边减去

1

u^{2} + 1 - 1 = 0 - 1

化简

u^{2} = - 1

u^{2} = - 1

对于

x^{2} = f (a)

解为

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

u = - 1, u = - - 1

化简

- 1 : i

- 1

使用虚数运算法则:

- 1 = i

= i

化简

- - 1 : - i

- - 1

使用虚数运算法则:

- 1 = i

= - i

u = i, u = - i

u = i, u = - i

u = cot (x)

代回

cot (x) = i, cot (x) = - i

cot (x) = i, cot (x) = - i

cot (x) = i :

无解

cot (x) = i

无解

cot (x) = - i :

无解

cot (x) = - i

无解

合并所有解

无解

cot (x) - tan (x) = 0 : x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

cot (x) - tan (x) = 0

使用三角恒等式改写

cot (x) - tan (x)

使用基本三角恒等式:

tan (x) = \frac{1}{cot ( x )}

= cot (x) - \frac{1}{cot ( x )}

cot (x) - \frac{1}{cot ( x )} = 0

用替代法求解

cot (x) - \frac{1}{cot ( x )} = 0

令

: cot (x) = u

u - \frac{1}{u} = 0

u - \frac{1}{u} = 0 : u = 1, u = - 1

u - \frac{1}{u} = 0

在两边乘以

u

u - \frac{1}{u} = 0

在两边乘以

u

uu - \frac{1}{u} u = 0 \cdot u

化简

uu - \frac{1}{u} u = 0 \cdot u

化简

uu : u^{2}

uu

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= u^{1 + 1}

数字相加:

1 + 1 = 2

= u^{2}

化简

- \frac{1}{u} u : - 1

- \frac{1}{u} u

分式相乘:

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

= - \frac{1 \cdot u}{u}

约分:

u

= - 1

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

u^{2} - 1 = 0

u^{2} - 1 = 0

u^{2} - 1 = 0

解

u^{2} - 1 = 0 : u = 1, u = - 1

u^{2} - 1 = 0

将

1

到右边

u^{2} - 1 = 0

两边加上

1

u^{2} - 1 + 1 = 0 + 1

化简

u^{2} = 1

u^{2} = 1

对于

x^{2} = f (a)

解为

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

u = 1, u = - 1

1 = 1

1

使用法则

1 = 1

= 1

- 1 = - 1

- 1

使用法则

1 = 1

= - 1

u = 1, u = - 1

u = 1, u = - 1

验证解

找到无定义的点（奇点）:

u = 0

取

u - \frac{1}{u}

的分母，令其等于零

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = 1, u = - 1

u = cot (x)

代回

cot (x) = 1, cot (x) = - 1

cot (x) = 1, cot (x) = - 1

cot (x) = 1 : x = \frac{π}{4} + πn

cot (x) = 1

cot (x) = 1

的通解

cot (x)

周期表（周期为

πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

cot (x) = - 1 : x = \frac{3 π}{4} + πn

cot (x) = - 1

cot (x) = - 1

的通解

cot (x)

周期表（周期为

πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

合并所有解

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

合并所有解

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

作图

流行的例子

(cos^2(a)-1)/(sin^2(a)+1)=0

\frac{cos ^{2} ( a ) - 1}{sin ^{2} ( a ) + 1} = 0

7sin^2(x)+2sin^2(x)-3cos^2(x)=0

7 sin^{2} (x) + 2 sin^{2} (x) - 3 cos^{2} (x) = 0

sin^2(x)-cos^2(x)=cos^4(x)

sin^{2} (x) - cos^{2} (x) = cos^{4} (x)

sin^2(x)+2cos^2(x)=1

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

(sin^2(a)+1)/(tan^2(a))=1

\frac{sin ^{2} ( a ) + 1}{tan ^{2} ( a )} = 1