zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

证明 (sec(A))/(sin(A))-(sin(A))/(cos(A))=cot(A)

解答

证明

\frac{sec ( A )}{sin ( A )} - \frac{sin ( A )}{cos ( A )} = cot (A)

解答

真

求解步骤

\frac{sec ( A )}{sin ( A )} - \frac{sin ( A )}{cos ( A )} = cot (A)

调整左侧

\frac{sec ( A )}{sin ( A )} - \frac{sin ( A )}{cos ( A )}

用 sin, cos 表示

\frac{sec ( A )}{sin ( A )} - \frac{sin ( A )}{cos ( A )}

使用基本三角恒等式:

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

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

化简

\frac{\frac{1}{c o s ( A )}}{sin ( A )} - \frac{sin ( A )}{cos ( A )} : \frac{1 - sin ^{2} ( A )}{cos ( A ) sin ( A )}

\frac{\frac{1}{c o s ( A )}}{sin ( A )} - \frac{sin ( A )}{cos ( A )}

使用分式法则:

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

= \frac{1}{cos ( A ) sin ( A )} - \frac{sin ( A )}{cos ( A )}

cos (A) sin (A), cos (A)

的最小公倍数:

cos (A) sin (A)

cos (A) sin (A), cos (A)

最小公倍数 (LCM)

计算出由出现在

cos (A) sin (A)

或

cos (A)

中的因子组成的表达式

= cos (A) sin (A)

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

cos (A) sin (A)

对于

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

将分母和分子乘以

sin (A)

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

= \frac{1}{cos ( A ) sin ( A )} - \frac{sin ^{2} ( A )}{cos ( A ) sin ( A )}

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

= \frac{1 - sin ^{2} ( A )}{cos ( A ) sin ( A )}

= \frac{1 - sin ^{2} ( A )}{cos ( A ) sin ( A )}

使用三角恒等式改写

\frac{1 - sin ^{2} ( A )}{cos ( A ) sin ( A )}

使用毕达哥拉斯恒等式:

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

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

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

约分:

cos (A)

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

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

使用基本三角恒等式:

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

= cot (A)

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 csc(θ)=sec(θ)cot(θ)

p ro v e csc (θ) = sec (θ) cot (θ)

证明 cos(-2x)=cosh^2(x)+sinh^2(x)

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

证明 (cos^2(x)+sin^2(x))=1

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

证明 sec^2(a)= 1/(1-sin^2(a))

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

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

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