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受欢迎的三角函数 >

证明 (1-tan(x))/(1-cot(x))=-tan(x)

解答

证明

\frac{1 - tan ( x )}{1 - cot ( x )} = - tan (x)

解答

真

求解步骤

\frac{1 - tan ( x )}{1 - cot ( x )} = - tan (x)

调整左侧

\frac{1 - tan ( x )}{1 - cot ( x )}

用 sin, cos 表示

\frac{1 - tan ( x )}{1 - cot ( x )}

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

化简

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

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

将项转换为分式:

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

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

乘以:

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

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

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

化简

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

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

将项转换为分式:

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

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

乘以:

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

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

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

分式相除:

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

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

sin (x) - cos (x) = - (cos (x) - sin (x))

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

整理后得

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

约分:

cos (x) - sin (x)

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

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

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

使用基本三角恒等式:

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

= - tan (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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