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受欢迎的三角函数 >

证明 cot(x/2)=(sin(x))/(1-cos(x))

解答

证明

cot (\frac{x}{2}) = \frac{sin ( x )}{1 - cos ( x )}

解答

真

求解步骤

cot (\frac{x}{2}) = \frac{sin ( x )}{1 - cos ( x )}

令

: u = \frac{x}{2}

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

证明

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

真

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

调整右侧

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

使用三角恒等式改写

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

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

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

使用倍角公式:

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

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

化简

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

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

乘开

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

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

- (1 - 2 sin^{2} (u)) : - 1 + 2 sin^{2} (u)

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

打开括号

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

使用加减运算法则

- (- a) = a, - (a) = - a

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

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

1 - 1 = 0

= 2 sin^{2} (u)

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

数字相除:

\frac{2}{2} = 1

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

约分:

sin (u)

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

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

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

使用基本三角恒等式:

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

= cot (u)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

因此

cot (\frac{x}{2}) = \frac{sin ( x )}{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

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

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p ro v e sin (2 z) = 2 sin (z) cos (z)

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p ro v e sin^{5} (x) = (sin (x))^{5}