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受欢迎的三角函数 >

证明 1/(sec^3(x)cos^4(x))=sec(x)

解答

证明

\frac{1}{sec ^{3} ( x ) cos ^{4} ( x )} = sec (x)

解答

真

求解步骤

\frac{1}{sec ^{3} ( x ) cos ^{4} ( x )} = sec (x)

调整左侧

\frac{1}{sec ^{3} ( x ) cos ^{4} ( x )}

用 sin, cos 表示

\frac{1}{cos ^{4} ( x ) sec ^{3} ( x )}

使用基本三角恒等式:

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

= \frac{1}{cos ^{4} ( x ) ( \frac{1}{c o s ( x )} ) ^{3}}

化简

\frac{1}{cos ^{4} ( x ) ( \frac{1}{c o s ( x )} ) ^{3}} : \frac{1}{cos ( x )}

\frac{1}{cos ^{4} ( x ) ( \frac{1}{c o s ( x )} ) ^{3}}

(\frac{1}{cos ( x )})^{3} = \frac{1}{cos ^{3} ( x )}

(\frac{1}{cos ( x )})^{3}

使用指数法则:

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

= \frac{1 ^{3}}{cos ^{3} ( x )}

使用法则

1^{a} = 1

1^{3} = 1

= \frac{1}{cos ^{3} ( x )}

= \frac{1}{\frac{1}{c o s ^{3} ( x )} cos ^{4} ( x )}

乘

cos^{4} (x) \frac{1}{cos ^{3} ( x )} : cos (x)

cos^{4} (x) \frac{1}{cos ^{3} ( x )}

分式相乘:

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

= \frac{1 \cdot cos ^{4} ( x )}{cos ^{3} ( x )}

乘以:

1 \cdot cos^{4} (x) = cos^{4} (x)

= \frac{cos ^{4} ( x )}{cos ^{3} ( x )}

使用指数法则:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{cos ^{4} ( x )}{cos ^{3} ( x )} = cos^{4 - 3} (x)

= cos^{4 - 3} (x)

数字相减:

4 - 3 = 1

= 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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