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(x)csc(x)=tan(x)sin(x)+cos(x)

解答

证明

tan (x) csc (x) = tan (x) sin (x) + cos (x)

解答

真

求解步骤

tan (x) csc (x) = tan (x) sin (x) + cos (x)

调整左侧

tan (x) csc (x)

用 sin, cos 表示

csc (x) tan (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

\frac{1}{sin ( x )} \cdot \frac{sin ( x )}{cos ( x )} : \frac{1}{cos ( x )}

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

分式相乘:

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

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

约分:

sin (x)

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

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

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

调整右侧

tan (x) sin (x) + cos (x)

用 sin, cos 表示

cos (x) + sin (x) tan (x)

使用基本三角恒等式:

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

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

化简

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

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

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

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

分式相乘:

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

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

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

sin (x) sin (x)

使用指数法则:

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

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

= sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= sin^{2} (x)

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

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

将项转换为分式:

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

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

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

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

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

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

cos (x) cos (x)

使用指数法则:

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

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

= cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= cos^{2} (x)

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

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

= \frac{1}{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 (cos((5*pi/3)/4))=(cos(5 pi/(12)))

p ro v e (cos (\frac{5 \cdot \frac{π}{3}}{4})) = (cos (5 \frac{π}{12}))

证明 tan(21)=2tan(t)

p ro v e tan (2 1^{\circ}) = 2 tan (t)

证明 tan^2(x)+sin^2(x)=1

p ro v e tan^{2} (x) + sin^{2} (x) = 1

证明 sin^2(θ)+cos^2(θ)=sec^2(θ)

p ro v e sin^{2} (θ) + cos^{2} (θ) = sec^{2} (θ)

证明 sin(x)-cos(x)+1=0

p ro v e sin (x) - cos (x) + 1 = 0