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受欢迎的三角函数 >

1/(cot(x))-(sec(x))/(csc(x))=cos(x)

解答

\frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} = cos (x)

解答

x \in R 无解

求解步骤

\frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} = cos (x)

两边减去

cos (x)

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

化简

\frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} - cos (x) : \frac{csc ( x ) - sec ( x ) cot ( x ) - cos ( x ) cot ( x ) csc ( x )}{cot ( x ) csc ( x )}

\frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} - cos (x)

将项转换为分式:

cos (x) = \frac{cos ( x )}{1}

= \frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} - \frac{cos ( x )}{1}

cot (x), csc (x), 1

的最小公倍数:

cot (x) csc (x)

cot (x), csc (x), 1

最小公倍数 (LCM)

计算出由至少在以下一个因式表达式中出现的因子组成的表达式

= cot (x) csc (x)

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

cot (x) csc (x)

对于

\frac{1}{cot ( x )} :

将分母和分子乘以

csc (x)

\frac{1}{cot ( x )} = \frac{1 \cdot csc ( x )}{cot ( x ) csc ( x )} = \frac{csc ( x )}{cot ( x ) csc ( x )}

对于

\frac{sec ( x )}{csc ( x )} :

将分母和分子乘以

cot (x)

\frac{sec ( x )}{csc ( x )} = \frac{sec ( x ) cot ( x )}{csc ( x ) cot ( x )}

对于

\frac{cos ( x )}{1} :

将分母和分子乘以

cot (x) csc (x)

\frac{cos ( x )}{1} = \frac{cos ( x ) cot ( x ) csc ( x )}{1 \cdot cot ( x ) csc ( x )} = \frac{cos ( x ) cot ( x ) csc ( x )}{cot ( x ) csc ( x )}

= \frac{csc ( x )}{cot ( x ) csc ( x )} - \frac{sec ( x ) cot ( x )}{csc ( x ) cot ( x )} - \frac{cos ( x ) cot ( x ) csc ( x )}{cot ( x ) csc ( x )}

因为分母相等，所以合并分式:

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

= \frac{csc ( x ) - sec ( x ) cot ( x ) - cos ( x ) cot ( x ) csc ( x )}{cot ( x ) csc ( x )}

\frac{csc ( x ) - sec ( x ) cot ( x ) - cos ( x ) cot ( x ) csc ( x )}{cot ( x ) csc ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

csc (x) - sec (x) cot (x) - cos (x) cot (x) csc (x) = 0

用 sin, cos 表示

csc (x) - cot (x) sec (x) - cos (x) cot (x) csc (x)

使用基本三角恒等式:

csc (x) = \frac{1}{sin ( x )}

= \frac{1}{sin ( x )} - cot (x) sec (x) - cos (x) cot (x) \frac{1}{sin ( x )}

使用基本三角恒等式:

cot (x) = \frac{cos ( x )}{sin ( x )}

= \frac{1}{sin ( x )} - \frac{cos ( x )}{sin ( x )} sec (x) - cos (x) \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )}

使用基本三角恒等式:

sec (x) = \frac{1}{cos ( x )}

= \frac{1}{sin ( x )} - \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} - cos (x) \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )}

化简

\frac{1}{sin ( x )} - \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} - cos (x) \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )} : - \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

\frac{1}{sin ( x )} - \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} - cos (x) \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )}

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} = \frac{1}{sin ( x )}

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )}

分式相乘:

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

= \frac{cos ( x ) \cdot 1}{sin ( x ) cos ( x )}

约分:

cos (x)

= \frac{1}{sin ( x )}

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

cos (x) \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )}

分式相乘:

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

= \frac{cos ( x ) \cdot 1 \cdot cos ( x )}{sin ( x ) sin ( x )}

cos (x) \cdot 1 \cdot cos (x) = cos^{2} (x)

cos (x) \cdot 1 \cdot cos (x)

使用指数法则:

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

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

= 1 \cdot cos^{1 + 1} (x)

整理后得

= cos^{2} (x)

= \frac{cos ^{2} ( x )}{sin ( x ) sin ( x )}

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

使用指数法则:

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

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

= sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= sin^{2} (x)

= \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

= \frac{1}{sin ( x )} - \frac{1}{sin ( x )} - \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

同类项相加:

1 \cdot \frac{1}{sin ( x )} - 1 \cdot \frac{1}{sin ( x )} = 0

= - \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

= - \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

- \frac{cos ^{2} ( x )}{sin ^{2} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- cos^{2} (x) = 0

两边除以

- 1

- cos^{2} (x) = 0

两边除以

- 1

- cos^{2} (x) = 0

两边除以

- 1

\frac{- cos ^{2} ( x )}{- 1} = \frac{0}{- 1}

化简

cos^{2} (x) = 0

cos^{2} (x) = 0

使用法则

x^{n} = 0 \Rightarrow x = 0

cos (x) = 0

cos (x) = 0

的通解

cos (x)

周期表（周期为

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

因为方程对以下值无定义:

\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn

x \in R 无解

作图

流行的例子

sin(x)=1,2pi<= x<= 4pi

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

(sin(x))/(cos(x))=0