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受欢迎的三角函数 >

证明 tan(u)+cot(u)=sec(u)csc(u)

解答

证明

tan (u) + cot (u) = sec (u) csc (u)

解答

真

求解步骤

tan (u) + cot (u) = sec (u) csc (u)

调整左侧

tan (u) + cot (u)

用 sin, cos 表示

cot (u) + tan (u)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

sin (u), cos (u)

的最小公倍数:

sin (u) cos (u)

sin (u), cos (u)

最小公倍数 (LCM)

计算出由出现在

sin (u)

或

cos (u)

中的因子组成的表达式

= sin (u) cos (u)

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 (u) cos (u)

对于

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

将分母和分子乘以

cos (u)

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

对于

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

将分母和分子乘以

sin (u)

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

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

使用三角恒等式改写

使用基本三角恒等式:

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

\frac{1}{cos ( u ) \frac{1}{c s c ( u )}}

使用基本三角恒等式:

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

\frac{1}{\frac{1}{s e c ( u )} \cdot \frac{1}{c s c ( u )}}

化简

\frac{1}{\frac{1}{s e c ( u )} \cdot \frac{1}{c s c ( u )}}

乘

\frac{1}{sec ( u )} \cdot \frac{1}{csc ( u )} : \frac{1}{sec ( u ) csc ( u )}

\frac{1}{sec ( u )} \cdot \frac{1}{csc ( u )}

分式相乘:

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

= \frac{1 \cdot 1}{sec ( u ) csc ( u )}

数字相乘:

1 \cdot 1 = 1

= \frac{1}{sec ( u ) csc ( u )}

= \frac{1}{\frac{1}{s e c ( u ) c s c ( u )}}

使用分式法则:

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

= \frac{sec ( u ) csc ( u )}{1}

使用法则

\frac{a}{1} = a

= sec (u) csc (u)

sec (u) csc (u)

sec (u) csc (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 cos(pi/(12))=(sqrt(2+\sqrt{3)})/2

p ro v e cos (\frac{π}{12}) = \frac{2 + 3}{2}

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

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p ro v e cot (\frac{π}{2} - θ) = tan (θ)

证明 (1-cos^2(x))(csc(x))=sin(x)

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

证明 (sin(x+y))/(sin(x)cos(y))=1+cot(x)tan(y)

p ro v e \frac{sin ( x + y )}{sin ( x ) cos ( y )} = 1 + cot (x) tan (y)