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受欢迎的三角函数 >

证明 (sec^4(α)-1)/(tan^2(α))=sec^2(α)+1

解答

证明

\frac{sec ^{4} ( α ) - 1}{tan ^{2} ( α )} = sec^{2} (α) + 1

解答

真

求解步骤

\frac{sec ^{4} ( α ) - 1}{tan ^{2} ( α )} = sec^{2} (α) + 1

调整左侧

\frac{sec ^{4} ( α ) - 1}{tan ^{2} ( α )}

使用三角恒等式改写

\frac{sec ^{4} ( α ) - 1}{tan ^{2} ( α )}

使用毕达哥拉斯恒等式:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= \frac{sec ^{4} ( α ) - 1}{sec ^{2} ( α ) - 1}

化简

\frac{sec ^{4} ( α ) - 1}{sec ^{2} ( α ) - 1} : sec^{2} (α) + 1

\frac{sec ^{4} ( α ) - 1}{sec ^{2} ( α ) - 1}

分解

sec^{4} (α) - 1 : (sec^{2} (α) + 1) (sec (α) + 1) (sec (α) - 1)

sec^{4} (α) - 1

将

sec^{4} (α) - 1

改写为

(sec^{2} (α))^{2} - 1^{2}

sec^{4} (α) - 1

将

1

改写为

1^{2}

= sec^{4} (α) - 1^{2}

使用指数法则:

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

sec^{4} (α) = (sec^{2} (α))^{2}

= (sec^{2} (α))^{2} - 1^{2}

= (sec^{2} (α))^{2} - 1^{2}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

(sec^{2} (α))^{2} - 1^{2} = (sec^{2} (α) + 1) (sec^{2} (α) - 1)

= (sec^{2} (α) + 1) (sec^{2} (α) - 1)

分解

sec^{2} (α) - 1 : (sec (α) + 1) (sec (α) - 1)

sec^{2} (α) - 1

将

1

改写为

1^{2}

= sec^{2} (α) - 1^{2}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

sec^{2} (α) - 1^{2} = (sec (α) + 1) (sec (α) - 1)

= (sec (α) + 1) (sec (α) - 1)

= (sec^{2} (α) + 1) (sec (α) + 1) (sec (α) - 1)

= \frac{( sec ^{2} ( α ) + 1 ) ( sec ( α ) + 1 ) ( sec ( α ) - 1 )}{sec ^{2} ( α ) - 1}

分解

sec^{2} (α) - 1 : (sec (α) + 1) (sec (α) - 1)

sec^{2} (α) - 1

将

1

改写为

1^{2}

= sec^{2} (α) - 1^{2}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

sec^{2} (α) - 1^{2} = (sec (α) + 1) (sec (α) - 1)

= (sec (α) + 1) (sec (α) - 1)

= \frac{( sec ^{2} ( α ) + 1 ) ( sec ( α ) + 1 ) ( sec ( α ) - 1 )}{( sec ( α ) + 1 ) ( sec ( α ) - 1 )}

消掉

\frac{( sec ^{2} ( α ) + 1 ) ( sec ( α ) + 1 ) ( sec ( α ) - 1 )}{( sec ( α ) + 1 ) ( sec ( α ) - 1 )} : sec^{2} (α) + 1

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

约分:

sec (α) + 1

= \frac{( sec ^{2} ( α ) + 1 ) ( sec ( α ) - 1 )}{sec ( α ) - 1}

约分:

sec (α) - 1

= sec^{2} (α) + 1

= sec^{2} (α) + 1

= sec^{2} (α) + 1

= sec^{2} (α) + 1

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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