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受欢迎的三角函数 >

证明 (sec(α)+csc(α))(cos(α)-sin(α))=cot(α)-tan(α)

解答

证明

(sec (α) + csc (α)) (cos (α) - sin (α)) = cot (α) - tan (α)

解答

真

求解步骤

(sec (α) + csc (α)) (cos (α) - sin (α)) = cot (α) - tan (α)

调整左侧

(sec (α) + csc (α)) (cos (α) - sin (α))

用 sin, cos 表示

(cos (α) - sin (α)) (csc (α) + sec (α))

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

化简

\frac{1}{sin ( α )} + \frac{1}{cos ( α )} : \frac{cos ( α ) + sin ( α )}{sin ( α ) cos ( α )}

\frac{1}{sin ( α )} + \frac{1}{cos ( α )}

sin (α), cos (α)

的最小公倍数:

sin (α) cos (α)

sin (α), cos (α)

最小公倍数 (LCM)

计算出由出现在

sin (α)

或

cos (α)

中的因子组成的表达式

= sin (α) cos (α)

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 (α) cos (α)

对于

\frac{1}{sin ( α )} :

将分母和分子乘以

cos (α)

\frac{1}{sin ( α )} = \frac{1 \cdot cos ( α )}{sin ( α ) cos ( α )} = \frac{cos ( α )}{sin ( α ) cos ( α )}

对于

\frac{1}{cos ( α )} :

将分母和分子乘以

sin (α)

\frac{1}{cos ( α )} = \frac{1 \cdot sin ( α )}{cos ( α ) sin ( α )} = \frac{sin ( α )}{sin ( α ) cos ( α )}

= \frac{cos ( α )}{sin ( α ) cos ( α )} + \frac{sin ( α )}{sin ( α ) cos ( α )}

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 ( α ) + sin ( α )}{sin ( α ) cos ( α )}

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

分式相乘:

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

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

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

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

乘开

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

(cos (α) + sin (α)) (cos (α) - sin (α))

使用平方差公式:

(a + b) (a - b) = a^{2} - b^{2}

a = cos (α), b = sin (α)

= cos^{2} (α) - sin^{2} (α)

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

调整右侧

cot (α) - tan (α)

用 sin, cos 表示

cot (α) - tan (α)

使用基本三角恒等式:

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

= \frac{cos ( α )}{sin ( α )} - tan (α)

使用基本三角恒等式:

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

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

化简

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

\frac{cos ( α )}{sin ( α )} - \frac{sin ( α )}{cos ( α )}

sin (α), cos (α)

的最小公倍数:

sin (α) cos (α)

sin (α), cos (α)

最小公倍数 (LCM)

计算出由出现在

sin (α)

或

cos (α)

中的因子组成的表达式

= sin (α) cos (α)

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 (α) cos (α)

对于

\frac{cos ( α )}{sin ( α )} :

将分母和分子乘以

cos (α)

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

对于

\frac{sin ( α )}{cos ( α )} :

将分母和分子乘以

sin (α)

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

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

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} ( α ) - sin ^{2} ( α )}{sin ( α ) cos ( α )}

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

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

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 1-(-cos(x))^2=1+cos^2(x)

p ro v e 1 - (- cos (x))^{2} = 1 + cos^{2} (x)

证明 csc^2(x/2)=(2sec(x))/(sec(x)-1)

p ro v e csc^{2} (\frac{x}{2}) = \frac{2 sec ( x )}{sec ( x ) - 1}

证明 (sec(x))/(-csc(x))+(-sin(x))/(cos(x))=-2tan(x)

p ro v e \frac{sec ( x )}{- csc ( x )} + \frac{- sin ( x )}{cos ( x )} = - 2 tan (x)

证明 1=(sec(u)+tan(u))/(sec(u)+tan(u))

p ro v e 1 = \frac{sec ( u ) + tan ( u )}{sec ( u ) + tan ( u )}

证明 (sin^2(x))/1*1/(sin^2(x))=1

p ro v e \frac{sin ^{2} ( x )}{1} \cdot \frac{1}{sin ^{2} ( x )} = 1