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 sin(x)*cos(x)(tan(x)+cot(x))=1

解答

证明

sin (x) \cdot cos (x) (tan (x) + cot (x)) = 1

解答

真

求解步骤

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

调整左侧

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

化简

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

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

sin (x), cos (x)

的最小公倍数:

sin (x) cos (x)

sin (x), cos (x)

最小公倍数 (LCM)

计算出由出现在

sin (x)

或

cos (x)

中的因子组成的表达式

= sin (x) cos (x)

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

sin (x) cos (x)

对于

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

将分母和分子乘以

cos (x)

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

对于

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

将分母和分子乘以

sin (x)

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

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

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

分式相乘:

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

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

约分:

cos (x)

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

约分:

sin (x)

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

证明 1/(1+tan^2(x))-1/(1+cot^2(x))=1

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

证明 csc^2(x)-1=(csc(x)-1)(csc(x)+1)

p ro v e csc^{2} (x) - 1 = (csc (x) - 1) (csc (x) + 1)

证明 csc^4(x)(cot^2(x))=csc^3(x)

p ro v e csc^{4} (x) (cot^{2} (x)) = csc^{3} (x)

证明 cos(x^2)= 1/(sec(x^2))

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