zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

证明 sec^2(B)-csc^2(B)=(tan(B)-cot(B))/(sin(B)cos(B))

解答

证明

sec^{2} (B) - csc^{2} (B) = \frac{tan ( B ) - cot ( B )}{sin ( B ) cos ( B )}

解答

真

求解步骤

sec^{2} (B) - csc^{2} (B) = \frac{tan ( B ) - cot ( B )}{sin ( B ) cos ( B )}

调整右侧

\frac{tan ( B ) - cot ( B )}{sin ( B ) cos ( B )}

用 sin, cos 表示

\frac{- cot ( B ) + tan ( B )}{cos ( B ) sin ( B )}

使用基本三角恒等式:

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

= \frac{- \frac{c o s ( B )}{s i n ( B )} + tan ( B )}{cos ( B ) sin ( B )}

使用基本三角恒等式:

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

= \frac{- \frac{c o s ( B )}{s i n ( B )} + \frac{s i n ( B )}{c o s ( B )}}{cos ( B ) sin ( B )}

化简

\frac{- \frac{c o s ( B )}{s i n ( B )} + \frac{s i n ( B )}{c o s ( B )}}{cos ( B ) sin ( B )} : \frac{- cos ^{2} ( B ) + sin ^{2} ( B )}{cos ^{2} ( B ) sin ^{2} ( B )}

\frac{- \frac{c o s ( B )}{s i n ( B )} + \frac{s i n ( B )}{c o s ( B )}}{cos ( B ) sin ( B )}

化简

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

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

sin (B), cos (B)

的最小公倍数:

sin (B) cos (B)

sin (B), cos (B)

最小公倍数 (LCM)

计算出由出现在

sin (B)

或

cos (B)

中的因子组成的表达式

= sin (B) cos (B)

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

sin (B) cos (B)

对于

\frac{cos ( B )}{sin ( B )} :

将分母和分子乘以

cos (B)

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

对于

\frac{sin ( B )}{cos ( B )} :

将分母和分子乘以

sin (B)

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

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

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

= \frac{\frac{- c o s ^{2} ( B ) + s i n ^{2} ( B )}{s i n ( B ) c o s ( B )}}{cos ( B ) sin ( B )}

使用分式法则:

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

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

sin (B) cos (B) cos (B) sin (B) = cos^{2} (B) sin^{2} (B)

sin (B) cos (B) cos (B) sin (B)

使用指数法则:

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

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

= sin (B) cos^{1 + 1} (B) sin (B)

数字相加:

1 + 1 = 2

= sin (B) cos^{2} (B) sin (B)

使用指数法则:

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

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

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

数字相加:

1 + 1 = 2

= cos^{2} (B) sin^{2} (B)

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

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

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

使用三角恒等式改写

使用基本三角恒等式:

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

\frac{- cos ^{2} ( B ) + ( \frac{1}{c s c ( B )} ) ^{2}}{cos ^{2} ( B ) ( \frac{1}{c s c ( B )} ) ^{2}}

使用基本三角恒等式:

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

\frac{- ( \frac{1}{s e c ( B )} ) ^{2} + ( \frac{1}{c s c ( B )} ) ^{2}}{( \frac{1}{s e c ( B )} ) ^{2} ( \frac{1}{c s c ( B )} ) ^{2}}

化简

\frac{- ( \frac{1}{s e c ( B )} ) ^{2} + ( \frac{1}{c s c ( B )} ) ^{2}}{( \frac{1}{s e c ( B )} ) ^{2} ( \frac{1}{c s c ( B )} ) ^{2}}

(\frac{1}{sec ( B )})^{2} = \frac{1}{sec ^{2} ( B )}

(\frac{1}{sec ( B )})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{sec ^{2} ( B )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( B )}

= \frac{- ( \frac{1}{s e c ( B )} ) ^{2} + ( \frac{1}{c s c ( B )} ) ^{2}}{( \frac{1}{c s c ( B )} ) ^{2} \frac{1}{s e c ^{2} ( B )}}

(\frac{1}{csc ( B )})^{2} = \frac{1}{csc ^{2} ( B )}

(\frac{1}{csc ( B )})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{csc ^{2} ( B )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{csc ^{2} ( B )}

= \frac{- ( \frac{1}{s e c ( B )} ) ^{2} + ( \frac{1}{c s c ( B )} ) ^{2}}{\frac{1}{s e c ^{2} ( B )} \cdot \frac{1}{c s c ^{2} ( B )}}

(\frac{1}{sec ( B )})^{2} = \frac{1}{sec ^{2} ( B )}

(\frac{1}{sec ( B )})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{sec ^{2} ( B )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( B )}

(\frac{1}{csc ( B )})^{2} = \frac{1}{csc ^{2} ( B )}

(\frac{1}{csc ( B )})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{csc ^{2} ( B )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{csc ^{2} ( B )}

= \frac{- \frac{1}{s e c ^{2} ( B )} + \frac{1}{c s c ^{2} ( B )}}{\frac{1}{s e c ^{2} ( B )} \cdot \frac{1}{c s c ^{2} ( B )}}

乘

\frac{1}{sec ^{2} ( B )} \cdot \frac{1}{csc ^{2} ( B )} : \frac{1}{sec ^{2} ( B ) csc ^{2} ( B )}

\frac{1}{sec ^{2} ( B )} \cdot \frac{1}{csc ^{2} ( B )}

分式相乘:

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

= \frac{1 \cdot 1}{sec ^{2} ( B ) csc ^{2} ( B )}

数字相乘:

1 \cdot 1 = 1

= \frac{1}{sec ^{2} ( B ) csc ^{2} ( B )}

= \frac{- \frac{1}{s e c ^{2} ( B )} + \frac{1}{c s c ^{2} ( B )}}{\frac{1}{s e c ^{2} ( B ) c s c ^{2} ( B )}}

化简

- \frac{1}{sec ^{2} ( B )} + \frac{1}{csc ^{2} ( B )} : \frac{- csc ^{2} ( B ) + sec ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )}

- \frac{1}{sec ^{2} ( B )} + \frac{1}{csc ^{2} ( B )}

sec^{2} (B), csc^{2} (B)

的最小公倍数:

sec^{2} (B) csc^{2} (B)

sec^{2} (B), csc^{2} (B)

最小公倍数 (LCM)

计算出由出现在

sec^{2} (B)

或

csc^{2} (B)

中的因子组成的表达式

= sec^{2} (B) csc^{2} (B)

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

sec^{2} (B) csc^{2} (B)

对于

\frac{1}{sec ^{2} ( B )} :

将分母和分子乘以

csc^{2} (B)

\frac{1}{sec ^{2} ( B )} = \frac{1 \cdot csc ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )} = \frac{csc ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )}

对于

\frac{1}{csc ^{2} ( B )} :

将分母和分子乘以

sec^{2} (B)

\frac{1}{csc ^{2} ( B )} = \frac{1 \cdot sec ^{2} ( B )}{csc ^{2} ( B ) sec ^{2} ( B )} = \frac{sec ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )}

= - \frac{csc ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )} + \frac{sec ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )}

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 ^{2} ( B ) + sec ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )}

= \frac{\frac{- c s c ^{2} ( B ) + s e c ^{2} ( B )}{s e c ^{2} ( B ) c s c ^{2} ( B )}}{\frac{1}{s e c ^{2} ( B ) c s c ^{2} ( B )}}

分式相除:

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

= \frac{( - csc ^{2} ( B ) + sec ^{2} ( B ) ) sec ^{2} ( B ) csc ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B ) \cdot 1}

整理后得

= \frac{( - csc ^{2} ( B ) + sec ^{2} ( B ) ) sec ^{2} ( B ) csc ^{2} ( B )}{sec ^{2} ( B ) csc ^{2} ( B )}

约分:

sec^{2} (B)

= - \frac{( - csc ^{2} ( B ) + sec ^{2} ( B ) ) csc ^{2} ( B )}{csc ^{2} ( B )}

约分:

csc^{2} (B)

= - - csc^{2} (B) + sec^{2} (B)

- csc^{2} (B) + sec^{2} (B)

- csc^{2} (B) + sec^{2} (B)

= sec^{2} (B) - csc^{2} (B)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

流行的例子

证明 2+sec(a)cos(a)=3

p ro v e 2 + sec (a) cos (a) = 3

证明 5cos^2(θ)+2sin^2(θ)=3cos^2(θ)+2

p ro v e 5 cos^{2} (θ) + 2 sin^{2} (θ) = 3 cos^{2} (θ) + 2

证明 (sec(a))/(tan(a)+cot(a))=sin(a)

p ro v e \frac{sec ( a )}{tan ( a ) + cot ( a )} = sin (a)

证明 sin((9pi)/2+x)=cos(x)

p ro v e sin (\frac{9 π}{2} + x) = cos (x)

证明 cos^4(β)-sin^4(β)=cos(2β)

p ro v e cos^{4} (β) - sin^{4} (β) = cos (2 β)