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 cot^2(α)+1= 1/(sin^2(α))

解答

证明

cot^{2} (α) + 1 = \frac{1}{sin ^{2} ( α )}

解答

真

求解步骤

cot^{2} (α) + 1 = \frac{1}{sin ^{2} ( α )}

调整左侧

cot^{2} (α) + 1

用 sin, cos 表示

1 + cot^{2} (α)

使用基本三角恒等式:

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

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

化简

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

1 + (\frac{cos ( α )}{sin ( α )})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

将项转换为分式:

1 = \frac{1 sin ^{2} ( α )}{sin ^{2} ( α )}

= \frac{1 \cdot sin ^{2} ( α )}{sin ^{2} ( α )} + \frac{cos ^{2} ( α )}{sin ^{2} ( 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{1 \cdot sin ^{2} ( α ) + cos ^{2} ( α )}{sin ^{2} ( α )}

乘以:

1 \cdot sin^{2} (α) = sin^{2} (α)

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

= \frac{1}{sin ^{2} ( α )}

= \frac{1}{sin ^{2} ( 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 sin(3x)=3sin(x)cos^2(x)-sin^3(x)

p ro v e sin (3 x) = 3 sin (x) cos^{2} (x) - sin^{3} (x)

证明 (cos(3x)-cos(5x))/(sin(3x)+sin(5x))=tan(x)

p ro v e \frac{cos ( 3 x ) - cos ( 5 x )}{sin ( 3 x ) + sin ( 5 x )} = tan (x)

证明 cos^2(x)= 1/2+1/2 cos(2x)

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

证明 tan(θ/2)=(1-cos(θ))/(sin(θ))

p ro v e tan (\frac{θ}{2}) = \frac{1 - cos ( θ )}{sin ( θ )}

证明 1/(2cos(x)sin(x))=csc(2x)

p ro v e \frac{1}{2 cos ( x ) sin ( x )} = csc (2 x)