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(a))/(tan(a)+cot(a))=sin(a)

解答

证明

\frac{sec ( a )}{tan ( a ) + cot ( a )} = sin (a)

解答

真

求解步骤

\frac{sec ( a )}{tan ( a ) + cot ( a )} = sin (a)

调整左侧

\frac{sec ( a )}{tan ( a ) + cot ( a )}

用 sin, cos 表示

\frac{sec ( a )}{cot ( a ) + tan ( a )}

使用基本三角恒等式:

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

= \frac{\frac{1}{c o s ( a )}}{cot ( a ) + tan ( a )}

使用基本三角恒等式:

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

= \frac{\frac{1}{c o s ( a )}}{\frac{c o s ( a )}{s i n ( a )} + tan ( a )}

使用基本三角恒等式:

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

= \frac{\frac{1}{c o s ( a )}}{\frac{c o s ( a )}{s i n ( a )} + \frac{s i n ( a )}{c o s ( a )}}

化简

\frac{\frac{1}{c o s ( a )}}{\frac{c o s ( a )}{s i n ( a )} + \frac{s i n ( a )}{c o s ( a )}} : \frac{sin ( a )}{cos ^{2} ( a ) + sin ^{2} ( a )}

\frac{\frac{1}{c o s ( a )}}{\frac{c o s ( a )}{s i n ( a )} + \frac{s i n ( a )}{c o s ( a )}}

使用分式法则:

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

= \frac{1}{cos ( a ) ( \frac{c o s ( a )}{s i n ( a )} + \frac{s i n ( a )}{c o s ( a )} )}

化简

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

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

sin (a), cos (a)

的最小公倍数:

sin (a) cos (a)

sin (a), cos (a)

最小公倍数 (LCM)

计算出由出现在

sin (a)

或

cos (a)

中的因子组成的表达式

= sin (a) cos (a)

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 (a) cos (a)

对于

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

将分母和分子乘以

cos (a)

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

对于

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

将分母和分子乘以

sin (a)

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

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

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

= \frac{1}{\frac{c o s ^{2} ( a ) + s i n ^{2} ( a )}{s i n ( a ) c o s ( a )} cos ( a )}

乘

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

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

分式相乘:

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

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

约分:

cos (a)

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

= \frac{1}{\frac{c o s ^{2} ( a ) + s i n ^{2} ( a )}{s i n ( a )}}

使用分式法则:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

= \frac{sin ( a )}{1}

使用法则

\frac{a}{1} = a

= sin (a)

= sin (a)

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((9pi)/2+x)=cos(x)

p ro v e sin (\frac{9 π}{2} + x) = cos (x)

证明 cos^4(β)-sin^4(β)=cos(2β)

p ro v e cos^{4} (β) - sin^{4} (β) = cos (2 β)

证明 cos^3(x)= 3/4 cos(x)+1/4 cos(3x)

p ro v e cos^{3} (x) = \frac{3}{4} cos (x) + \frac{1}{4} cos (3 x)

证明 tan(pi/4+x)=(1+sin(2x))/(cos(2x))

p ro v e tan (\frac{π}{4} + x) = \frac{1 + sin ( 2 x )}{cos ( 2 x )}

证明 sin^2(x)+1/(sec^2(x))=sin(x)csc(x)

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