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受欢迎的三角函数 >

证明 sec(x)cot(x)-cot(x)cos(x)=sin(x)

解答

证明

sec (x) cot (x) - cot (x) cos (x) = sin (x)

解答

真

求解步骤

sec (x) cot (x) - cot (x) cos (x) = sin (x)

调整左侧

sec (x) cot (x) - cot (x) cos (x)

用 sin, cos 表示

- cos (x) cot (x) + cot (x) sec (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

sec (x) = \frac{1}{cos ( x )}

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

化简

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

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

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

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

分式相乘:

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

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

约分:

cos (x)

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

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

使用法则

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

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

约分:

sin (x)

= sin (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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p ro v e sec (x) = sec (- x)