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 csc(x)(cos(x)+sin(x))=cot(x)+1

解答

证明

csc (x) (cos (x) + sin (x)) = cot (x) + 1

解答

真

求解步骤

csc (x) (cos (x) + sin (x)) = cot (x) + 1

调整右侧

cot (x) + 1

用 sin, cos 表示

1 + cot (x)

使用基本三角恒等式:

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

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

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

使用基本三角恒等式:

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

= (cos (x) + sin (x)) csc (x)

= csc (x) (cos (x) + sin (x))

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 cos^2(x)+cos^2(x)cot^2(x)=cot^2(x)

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

证明 sqrt(2)cos(x-pi/4)=cos(x)+sin(x)

p ro v e 2 cos (x - \frac{π}{4}) = cos (x) + sin (x)

证明 (cos(u))/(1-sin(u))=sec(u)+tan(u)

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

证明 cos(x-(3pi)/2)=-sin(x)

p ro v e cos (x - \frac{3 π}{2}) = - sin (x)

证明 sin^2(x)sec^2(x)+1=sec^2(x)

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