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(x)-tan(x))(1+sin(x))=cos(x)

解答

证明

(sec (x) - tan (x)) (1 + sin (x)) = cos (x)

解答

真

求解步骤

(sec (x) - tan (x)) (1 + sin (x)) = cos (x)

调整左侧

(sec (x) - tan (x)) (1 + sin (x))

用 sin, cos 表示

(1 + sin (x)) (sec (x) - tan (x))

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

(1 + sin (x)) (\frac{1}{cos ( x )} - \frac{sin ( x )}{cos ( x )}) : \frac{( 1 - sin ( x ) ) ( 1 + sin ( x ) )}{cos ( x )}

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

化简

\frac{1}{cos ( x )} - \frac{sin ( x )}{cos ( x )} : \frac{- sin ( x ) + 1}{cos ( x )}

\frac{1}{cos ( x )} - \frac{sin ( x )}{cos ( x )}

使用法则

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

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

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

分式相乘:

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

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

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

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

使用三角恒等式改写

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

乘开

(1 + sin (x)) (1 - sin (x)) : 1 - sin^{2} (x)

(1 + sin (x)) (1 - sin (x))

使用平方差公式:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = sin (x)

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

使用法则

1^{a} = 1

1^{2} = 1

= 1 - sin^{2} (x)

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

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} ( x )}{cos ( x )}

约分:

cos (x)

= cos (x)

= cos (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 tan(-x)=cos(x)

p ro v e tan (- x) = cos (x)

证明 cos(45-x)=cos(x-45)

p ro v e cos (4 5^{\circ} - x) = cos (x - 4 5^{\circ})

证明 (1-csc^2(A))/(csc^2(A))=-cos^2(A)

p ro v e \frac{1 - csc ^{2} ( A )}{csc ^{2} ( A )} = - cos^{2} (A)

证明 sin^{0.5}(x)cos(x)-sin^{2.5}(x)cos(x)=cos^3(x)sin(x)

p ro v e sin^{0.5} (x) cos (x) - sin^{2.5} (x) cos (x) = cos^{3} (x) sin (x)

证明 sin((2pi)/3)=(sqrt(3))/2

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