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受欢迎的三角函数 >

证明 cot(2x)=(csc^2(x)-2)/(2cot(x))

解答

证明

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

解答

真

求解步骤

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

调整右侧

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

= \frac{- 2 + ( \frac{1}{s i n ( x )} ) ^{2}}{2 cot ( x )}

使用基本三角恒等式:

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

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

化简

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

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

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

使用分式法则:

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

= \frac{( - 2 + \frac{1}{s i n ^{2} ( x )} ) sin ( x )}{cos ( x ) \cdot 2}

化简

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

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

将项转换为分式:

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

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

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

= \frac{\frac{- 2 s i n ^{2} ( x ) + 1}{s i n ^{2} ( x )} sin ( x )}{2 cos ( x )}

乘

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

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

分式相乘:

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

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

约分:

sin (x)

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

= \frac{\frac{- 2 s i n ^{2} ( x ) + 1}{s i n ( x )}}{2 cos ( x )}

使用分式法则:

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

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

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

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

使用三角恒等式改写

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

使用倍角公式:

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

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

使用倍角公式:

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

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

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

使用基本三角恒等式:

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

= cot (2 x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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p ro v e (tan (y) + cot (y)) sin (y) cos (y) = 1

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