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受欢迎的三角函数 >

证明 (csc(x)-cot(x))/(sec(x)-1)=cot(x)

解答

证明

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

解答

真

求解步骤

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

调整左侧

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

= \frac{- \frac{c o s ( x )}{s i n ( x )} + csc ( x )}{- 1 + sec ( x )}

使用基本三角恒等式:

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

= \frac{- \frac{c o s ( x )}{s i n ( x )} + \frac{1}{s i n ( x )}}{- 1 + sec ( x )}

使用基本三角恒等式:

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

= \frac{- \frac{c o s ( x )}{s i n ( x )} + \frac{1}{s i n ( x )}}{- 1 + \frac{1}{c o s ( x )}}

化简

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

\frac{- \frac{c o s ( x )}{s i n ( x )} + \frac{1}{s i n ( x )}}{- 1 + \frac{1}{c o s ( x )}}

合并分式

- \frac{cos ( x )}{sin ( x )} + \frac{1}{sin ( x )} : \frac{- cos ( x ) + 1}{sin ( x )}

使用法则

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

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

= \frac{\frac{- c o s ( x ) + 1}{s i n ( x )}}{- 1 + \frac{1}{c o s ( x )}}

使用分式法则:

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

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

化简

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

- 1 + \frac{1}{cos ( x )}

将项转换为分式:

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

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

乘以:

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

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

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

乘

sin (x) \frac{- cos ( x ) + 1}{cos ( x )} : \frac{sin ( x ) ( - cos ( x ) + 1 )}{cos ( x )}

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

分式相乘:

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

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

= \frac{- cos ( x ) + 1}{\frac{s i n ( x ) ( - c o s ( x ) + 1 )}{c o s ( x )}}

使用分式法则:

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

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

约分:

- cos (x) + 1

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

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

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

使用基本三角恒等式:

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

= cot (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

流行的例子

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