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受欢迎的三角函数 >

证明 2cot(2x)=cot(x)-tan(x)

解答

证明

2 cot (2 x) = cot (x) - tan (x)

解答

真

求解步骤

2 cot (2 x) = cot (x) - tan (x)

调整左侧

2 cot (2 x)

用 sin, cos 表示

2 cot (2 x)

使用基本三角恒等式:

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

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

化简

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

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

分式相乘:

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

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

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

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

使用三角恒等式改写

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

使用倍角公式:

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

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

数字相除:

\frac{2}{2} = 1

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

使用倍角公式:

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

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

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

调整右侧

cot (x) - tan (x)

用 sin, cos 表示

cot (x) - tan (x)

使用基本三角恒等式:

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

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

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 )}

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

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

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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