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受欢迎的三角函数 >

证明 cot(x)sin(x)sec^2(x)=sec(x)

解答

证明

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

解答

真

求解步骤

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

调整左侧

cot (x) sin (x) sec^{2} (x)

用 sin, cos 表示

cot (x) sec^{2} (x) sin (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

分式相乘:

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

= \frac{cos ( x ) ( \frac{1}{c o s ( x )} ) ^{2} sin ( x )}{sin ( x )}

约分:

sin (x)

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

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

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

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

使用法则

1^{a} = 1

1^{2} = 1

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

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

分式相乘:

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

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

乘以:

1 \cdot cos (x) = cos (x)

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

约分:

cos (x)

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

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

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

使用三角恒等式改写

使用基本三角恒等式:

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

\frac{1}{\frac{1}{s e c ( x )}}

化简

\frac{1}{\frac{1}{s e c ( x )}}

使用分式法则:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

使用法则

\frac{a}{1} = a

= sec (x)

sec (x)

sec (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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