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受欢迎的三角函数 >

证明 1/(cot(x)(1-cos(2x)))=csc(2x)

解答

证明

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

解答

真

求解步骤

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

调整左侧

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

使用三角恒等式改写

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

使用倍角公式:

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

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

乘开

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

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

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

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

打开括号

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

使用加减运算法则

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

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

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

1 - 1 = 0

= 2 sin^{2} (x)

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

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

用 sin, cos 表示

\frac{1}{2 cot ( x ) sin ^{2} ( x )}

使用基本三角恒等式:

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

= \frac{1}{2 \cdot \frac{c o s ( x )}{s i n ( x )} sin ^{2} ( x )}

化简

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

\frac{1}{2 \cdot \frac{c o s ( x )}{s i n ( x )} sin ^{2} ( x )}

乘

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

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

分式相乘:

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

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

约分:

sin (x)

= 2 cos (x) sin (x)

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

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

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

使用三角恒等式改写

\frac{1}{2 cos ( x ) sin ( x )}

使用倍角公式:

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

= \frac{1}{sin ( 2 x )}

= \frac{1}{sin ( 2 x )}

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}{\frac{1}{c s c ( 2 x )}}

化简

\frac{1}{\frac{1}{c s c ( 2 x )}}

使用分式法则:

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

= \frac{csc ( 2 x )}{1}

使用法则

\frac{a}{1} = a

= csc (2 x)

csc (2 x)

csc (2 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 (θ) + cot (θ)) cos (θ) = csc (θ)

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p ro v e cos^{4} (t) - sin^{4} (t) = 1 - 2 sin^{2} (t)

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p ro v e 1 + tan^{2} (- θ) = sec^{2} (θ)

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

证明 (cos(pi/2+x))/(cos(pi+x))=tan(x)

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