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受欢迎的三角函数 >

证明 (csc^2(x))/(cot(x))=sec(x)csc(x)

解答

证明

\frac{csc ^{2} ( x )}{cot ( x )} = sec (x) csc (x)

解答

真

求解步骤

\frac{csc ^{2} ( x )}{cot ( x )} = sec (x) csc (x)

调整左侧

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

= \frac{( \frac{1}{s i n ( x )} ) ^{2}}{cot ( x )}

使用基本三角恒等式:

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

= \frac{( \frac{1}{s i n ( x )} ) ^{2}}{\frac{c o s ( x )}{s i n ( x )}}

化简

\frac{( \frac{1}{s i n ( x )} ) ^{2}}{\frac{c o s ( x )}{s i n ( x )}} : \frac{1}{sin ( x ) cos ( x )}

\frac{( \frac{1}{s i n ( x )} ) ^{2}}{\frac{c o s ( x )}{s i n ( x )}}

使用分式法则:

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

= \frac{( \frac{1}{s i n ( x )} ) ^{2} sin ( x )}{cos ( x )}

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

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

使用指数法则:

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

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

使用法则

1^{a} = 1

1^{2} = 1

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

= \frac{\frac{1}{s i n ^{2} ( x )} sin ( x )}{cos ( x )}

乘

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

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

分式相乘:

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

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

乘以:

1 \cdot sin (x) = sin (x)

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

约分:

sin (x)

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

= \frac{\frac{1}{s i n ( x )}}{cos ( x )}

使用分式法则:

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

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

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

= \frac{1}{cos ( x ) sin ( 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}{cos ( x ) \frac{1}{c s c ( x )}}

使用基本三角恒等式:

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

\frac{1}{\frac{1}{s e c ( x )} \cdot \frac{1}{c s c ( x )}}

化简

\frac{1}{\frac{1}{s e c ( x )} \cdot \frac{1}{c s c ( x )}}

乘

\frac{1}{sec ( x )} \cdot \frac{1}{csc ( x )} : \frac{1}{sec ( x ) csc ( x )}

\frac{1}{sec ( x )} \cdot \frac{1}{csc ( x )}

分式相乘:

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

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

数字相乘:

1 \cdot 1 = 1

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

= \frac{1}{\frac{1}{s e c ( x ) c s c ( x )}}

使用分式法则:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

使用法则

\frac{a}{1} = a

= sec (x) csc (x)

sec (x) csc (x)

sec (x) csc (x)

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 (cos(A))/(1+sin(A))+(cos(A))/(1-sin(A))=2sec(A)

p ro v e \frac{cos ( A )}{1 + sin ( A )} + \frac{cos ( A )}{1 - sin ( A )} = 2 sec (A)

证明 cot(x/2)=(sin(x))/(1-cos(x))

p ro v e cot (\frac{x}{2}) = \frac{sin ( x )}{1 - cos ( x )}

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p ro v e - tan (y) + sec (y) = \frac{cos ( y )}{1 + sin ( y )}

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p ro v e sin (2 z) = 2 sin (z) cos (z)

证明 cot(x/2)=((1+cos(x)))/(sin(x))

p ro v e cot (\frac{x}{2}) = \frac{( 1 + cos ( x ) )}{sin ( x )}