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受欢迎的三角函数 >

证明 (sec^2(x)-1)/(sin(x))=(sin(x))/(1-sin^2(x))

解答

证明

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

解答

真

求解步骤

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

调整左侧

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

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

化简

\frac{- 1 + ( \frac{1}{c o s ( x )} ) ^{2}}{sin ( x )} : \frac{- cos ^{2} ( x ) + 1}{cos ^{2} ( x ) sin ( x )}

\frac{- 1 + ( \frac{1}{c o s ( x )} ) ^{2}}{sin ( x )}

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

化简

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

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

将项转换为分式:

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

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

乘以:

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

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

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

使用分式法则:

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

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

化简

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

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

乘开

1 - (1 - sin^{2} (x)) : sin^{2} (x)

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

- (1 - sin^{2} (x)) : - 1 + sin^{2} (x)

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

打开括号

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

使用加减运算法则

- (- a) = a, - (a) = - a

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

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

1 - 1 = 0

= sin^{2} (x)

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

约分:

sin (x)

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

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

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

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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