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受欢迎的三角函数 >

证明 (2cot(x))/(csc^2(x))=sin(2x)

解答

证明

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

解答

真

求解步骤

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

调整左侧

\frac{2 cot ( x )}{csc ^{2} ( x )}

用 sin, cos 表示

\frac{2 cot ( x )}{csc ^{2} ( x )}

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

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

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

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

使用法则

1^{a} = 1

1^{2} = 1

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

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

乘

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

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

分式相乘:

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

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

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

分式相除:

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

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

整理后得

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

约分:

sin (x)

= 2 cos (x) sin (x)

= 2 cos (x) sin (x)

= 2 cos (x) sin (x)

使用三角恒等式改写

2 cos (x) sin (x)

使用倍角公式:

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

= sin (2 x)

= sin (2 x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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