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 tan(u)(csc(u)-sin(u))=cos(u)

解答

证明

tan (u) (csc (u) - sin (u)) = cos (u)

解答

真

求解步骤

tan (u) (csc (u) - sin (u)) = cos (u)

调整左侧

tan (u) (csc (u) - sin (u))

用 sin, cos 表示

(csc (u) - sin (u)) tan (u)

使用基本三角恒等式:

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

= (\frac{1}{sin ( u )} - sin (u)) tan (u)

使用基本三角恒等式:

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

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

化简

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

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

分式相乘:

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

= \frac{sin ( u ) ( \frac{1}{s i n ( u )} - sin ( u ) )}{cos ( u )}

化简

\frac{1}{sin ( u )} - sin (u) : \frac{1 - sin ^{2} ( u )}{sin ( u )}

\frac{1}{sin ( u )} - sin (u)

将项转换为分式:

sin (u) = \frac{sin ( u ) sin ( u )}{sin ( u )}

= \frac{1}{sin ( u )} - \frac{sin ( u ) sin ( u )}{sin ( u )}

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 - sin ( u ) sin ( u )}{sin ( u )}

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

1 - sin (u) sin (u)

sin (u) sin (u) = sin^{2} (u)

sin (u) sin (u)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

sin (u) sin (u) = sin^{1 + 1} (u)

= sin^{1 + 1} (u)

数字相加:

1 + 1 = 2

= sin^{2} (u)

= 1 - sin^{2} (u)

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

= \frac{\frac{- s i n ^{2} ( u ) + 1}{s i n ( u )} sin ( u )}{cos ( u )}

乘

sin (u) \frac{1 - sin ^{2} ( u )}{sin ( u )} : 1 - sin^{2} (u)

sin (u) \frac{1 - sin ^{2} ( u )}{sin ( u )}

分式相乘:

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

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

约分:

sin (u)

= 1 - sin^{2} (u)

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

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

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

使用三角恒等式改写

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

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} ( u )}{cos ( u )}

约分:

cos (u)

= cos (u)

= cos (u)

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 1-2sin^2(x)+sin^4(x)=cos^4(x)

p ro v e 1 - 2 sin^{2} (x) + sin^{4} (x) = cos^{4} (x)

证明 (sin(3θ))/(sin(θ))=3-4sin^2(θ)

p ro v e \frac{sin ( 3 θ )}{sin ( θ )} = 3 - 4 sin^{2} (θ)

证明 tan(pi-θ)=tan(θ)

p ro v e tan (π - θ) = tan (θ)

证明 6cos^2(x)-3=3-6sin^2(x)

p ro v e 6 cos^{2} (x) - 3 = 3 - 6 sin^{2} (x)

证明 sin(x)tan(x)+cos(-x)=sec(x)

p ro v e sin (x) tan (x) + cos (- x) = sec (x)