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受欢迎的三角函数 >

证明 csc^2(x)-1=(cos(x))/(tan(x)sin(x))

解答

证明

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

解答

真

求解步骤

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

调整右侧

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

= \frac{cos ( x )}{sin ( x ) \frac{s i n ( x )}{c o s ( x )}}

化简

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

\frac{cos ( x )}{sin ( x ) \frac{s i n ( x )}{c o s ( x )}}

乘

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

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

分式相乘:

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

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

= \frac{cos ( x )}{\frac{s i n ^{2} ( x )}{c o s ( x )}}

使用分式法则:

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

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

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

cos (x) cos (x)

使用指数法则:

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

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

= cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= cos^{2} (x)

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

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

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{1 - ( \frac{1}{c s c ( x )} ) ^{2}}{( \frac{1}{c s c ( x )} ) ^{2}}

化简

\frac{1 - ( \frac{1}{c s c ( x )} ) ^{2}}{( \frac{1}{c s c ( x )} ) ^{2}}

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

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

使用指数法则:

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

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

使用法则

1^{a} = 1

1^{2} = 1

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

= \frac{1 - ( \frac{1}{c s c ( x )} ) ^{2}}{\frac{1}{c s c ^{2} ( x )}}

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

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

使用指数法则:

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

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

使用法则

1^{a} = 1

1^{2} = 1

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

= \frac{1 - \frac{1}{c s c ^{2} ( x )}}{\frac{1}{c s c ^{2} ( x )}}

使用分式法则:

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

= \frac{( 1 - \frac{1}{c s c ^{2} ( x )} ) csc ^{2} ( x )}{1}

化简

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

1 - \frac{1}{csc ^{2} ( x )}

将项转换为分式:

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

= \frac{1 \cdot csc ^{2} ( x )}{csc ^{2} ( x )} - \frac{1}{csc ^{2} ( 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{1 \cdot csc ^{2} ( x ) - 1}{csc ^{2} ( x )}

乘以:

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

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

= \frac{\frac{c s c ^{2} ( x ) - 1}{c s c ^{2} ( x )} csc ^{2} ( x )}{1}

使用分式法则:

\frac{a}{1} = a

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

分式相乘:

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

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

约分:

csc^{2} (x)

= csc^{2} (x) - 1

csc^{2} (x) - 1

csc^{2} (x) - 1

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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