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受欢迎的三角函数 >

证明 (1+tan(x))(1-cot(x))=tan(x)-cot(x)

解答

证明

(1 + tan (x)) (1 - cot (x)) = tan (x) - cot (x)

解答

真

求解步骤

(1 + tan (x)) (1 - cot (x)) = tan (x) - cot (x)

调整左侧

(1 + tan (x)) (1 - cot (x))

用 sin, cos 表示

(1 + tan (x)) (1 - cot (x))

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

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

分式相乘:

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

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

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

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

调整右侧

tan (x) - cot (x)

用 sin, cos 表示

- cot (x) + tan (x)

使用基本三角恒等式:

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

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

化简

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

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

sin (x), cos (x)

的最小公倍数:

sin (x) cos (x)

sin (x), cos (x)

最小公倍数 (LCM)

计算出由出现在

sin (x)

或

cos (x)

中的因子组成的表达式

= sin (x) cos (x)

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 (x) cos (x)

对于

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

将分母和分子乘以

cos (x)

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

对于

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

将分母和分子乘以

sin (x)

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

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

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

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

分解

- cos^{2} (x) + sin^{2} (x) : (sin (x) + cos (x)) (sin (x) - cos (x))

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

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

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

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

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

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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