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受欢迎的三角函数 >

证明 (cos(2x))/(sin(x))=cos(x)cot(x)-sin(x)

解答

证明

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

解答

真

求解步骤

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

调整左侧

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

使用三角恒等式改写

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

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

调整右侧

cos (x) cot (x) - sin (x)

用 sin, cos 表示

- sin (x) + cos (x) cot (x)

使用基本三角恒等式:

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

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

化简

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

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

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

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

分式相乘:

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

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

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

cos (x) cos (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

cos (x) cos (x) = cos^{1 + 1} (x)

= cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= cos^{2} (x)

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

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

将项转换为分式:

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

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

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

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

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

sin (x) sin (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

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

= sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= sin^{2} (x)

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

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

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

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

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

流行的例子

证明 cot^2(2x)+cos^2(2x)+sin^2(2x)=csc^2(2x)

p ro v e cot^{2} (2 x) + cos^{2} (2 x) + sin^{2} (2 x) = csc^{2} (2 x)

证明 25sec^2(5x)=25+(5tan(5x))^2

p ro v e 25 sec^{2} (5 x) = 25 + (5 tan (5 x))^{2}

证明 tan(x)=csc(2x)-cot(2x)

p ro v e tan (x) = csc (2 x) - cot (2 x)

证明 sin^2(-x)+cos^2(-x)=1

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

证明 (sec(x)-cos(x))/(tan(x))=sin(x)

p ro v e \frac{sec ( x ) - cos ( x )}{tan ( x )} = sin (x)