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受欢迎的三角函数 >

证明 cot(2x)=(1/2)(cot(x)-tan(x))

解答

证明

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

解答

真

求解步骤

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

调整右侧

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

分式相乘:

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

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

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

1 \cdot (\frac{cos ( x )}{sin ( x )} - \frac{sin ( x )}{cos ( x )})

乘以:

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

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

去除括号:

(a) = a

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

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

化简

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

\frac{cos ( x )}{sin ( x )} - \frac{sin ( x )}{cos ( x )}

sin (x), cos (x)

的最小公倍数:

sin (x) cos (x)

sin (x), cos (x)

最小公倍数 (LCM)

计算出由出现在

sin (x)

或

cos (x)

中的因子组成的表达式

= sin (x) cos (x)

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 (x) cos (x)

对于

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

将分母和分子乘以

cos (x)

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

对于

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

将分母和分子乘以

sin (x)

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

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

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

使用分式法则:

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

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

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

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

使用三角恒等式改写

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

使用倍角公式:

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

流行的例子

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p ro v e \frac{cos ^{2} ( x )}{1 + sin ( x )} = sin (x)

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p ro v e sin (x) + cos (x) = sin (x) cos (x)