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 (1-cos^2(θ))(1+cot^2(θ))=1

解答

证明

(1 - cos^{2} (θ)) (1 + cot^{2} (θ)) = 1

解答

真

求解步骤

(1 - cos^{2} (θ)) (1 + cot^{2} (θ)) = 1

调整左侧

(1 - cos^{2} (θ)) (1 + cot^{2} (θ))

用 sin, cos 表示

(1 + cot^{2} (θ)) (1 - cos^{2} (θ))

使用基本三角恒等式:

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

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

化简

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

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

使用指数法则:

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

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

化简

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

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

将项转换为分式:

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

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

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} ( θ ) + cos ^{2} ( θ )}{sin ^{2} ( θ )}

乘以:

1 \cdot sin^{2} (θ) = sin^{2} (θ)

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

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

分式相乘:

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

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

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

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

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

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

化简

cos^{2} (θ) + 1 - cos^{2} (θ) : 1

cos^{2} (θ) + 1 - cos^{2} (θ)

对同类项分组

= cos^{2} (θ) - cos^{2} (θ) + 1

同类项相加:

cos^{2} (θ) - cos^{2} (θ) = 0

= 1

= 1 \cdot (- cos^{2} (θ) + 1)

乘以:

(1 - cos^{2} (θ)) \cdot 1 = (1 - cos^{2} (θ))

= (1 - cos^{2} (θ))

去除括号:

(a) = a

= 1 - cos^{2} (θ)

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

使用法则

\frac{a}{a} = 1

= 1

= 1

= 1

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 cot(x)sin(x)=cos(x)

p ro v e cot (x) sin (x) = cos (x)

证明 sin(2x)+cos(2x)-1=2sin(x)(cos(x)-sin(x))

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

证明 (csc^2(x)-1)sin(x)=cot(x)cos(x)

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

证明 cot(x)(1-cos^2(x))= 1/2 sin(2x)

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

证明 (sec^2(t))/(tan(t))=sec(t)csc(t)

p ro v e \frac{sec ^{2} ( t )}{tan ( t )} = sec (t) csc (t)