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(2A)+cot(2A)=cot(A)

解答

证明

csc (2 A) + cot (2 A) = cot (A)

解答

真

求解步骤

csc (2 A) + cot (2 A) = cot (A)

调整左侧

csc (2 A) + cot (2 A)

用 sin, cos 表示

cot (2 A) + csc (2 A)

使用基本三角恒等式:

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

= \frac{cos ( 2 A )}{sin ( 2 A )} + csc (2 A)

使用基本三角恒等式:

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

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

化简

\frac{cos ( 2 A )}{sin ( 2 A )} + \frac{1}{sin ( 2 A )} : \frac{cos ( 2 A ) + 1}{sin ( 2 A )}

\frac{cos ( 2 A )}{sin ( 2 A )} + \frac{1}{sin ( 2 A )}

使用法则

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

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

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

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

使用三角恒等式改写

\frac{1 + cos ( 2 A )}{sin ( 2 A )}

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

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

使用倍角公式:

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

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

化简

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

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

1 + 2 cos^{2} (A) - 1 = 2 cos^{2} (A)

1 + 2 cos^{2} (A) - 1

对同类项分组

= 2 cos^{2} (A) + 1 - 1

1 - 1 = 0

= 2 cos^{2} (A)

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

数字相除:

\frac{2}{2} = 1

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

约分:

cos (A)

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

p ro v e sin (x) - cos (x) = 1 - sin (2 x)

证明 9sec^2(x)-5tan^2(x)=5+4sec^2(x)

p ro v e 9 sec^{2} (x) - 5 tan^{2} (x) = 5 + 4 sec^{2} (x)

证明 sin(3A)=3sin(A)-4sin^3(A)

p ro v e sin (3 A) = 3 sin (A) - 4 sin^{3} (A)

证明 cot^4(x)+cot^2(x)=cot^2(x)csc^2(x)

p ro v e cot^{4} (x) + cot^{2} (x) = cot^{2} (x) csc^{2} (x)

证明 cot^3(x)=cot(x)(csc^2(x)-1)

p ro v e cot^{3} (x) = cot (x) (csc^{2} (x) - 1)