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受欢迎的三角函数 >

证明 tan(x)sec^4(x)=(sin(x))/(cos^5(x))

解答

证明

tan (x) sec^{4} (x) = \frac{sin ( x )}{cos ^{5} ( x )}

解答

真

求解步骤

tan (x) sec^{4} (x) = \frac{sin ( x )}{cos ^{5} ( x )}

调整左侧

tan (x) sec^{4} (x)

用 sin, cos 表示

sec^{4} (x) tan (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

(\frac{1}{cos ( x )})^{4} \frac{sin ( x )}{cos ( x )} : \frac{sin ( x )}{cos ^{5} ( x )}

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

分式相乘:

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

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

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

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

使用指数法则:

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

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

使用法则

1^{a} = 1

1^{4} = 1

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

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

乘

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

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

分式相乘:

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

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

乘以:

1 \cdot sin (x) = sin (x)

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

= \frac{\frac{s i n ( x )}{c o s ^{4} ( x )}}{cos ( x )}

使用分式法则:

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

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

cos^{4} (x) cos (x) = cos^{5} (x)

cos^{4} (x) cos (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

cos^{4} (x) cos (x) = cos^{4 + 1} (x)

= cos^{4 + 1} (x)

数字相加:

4 + 1 = 5

= cos^{5} (x)

= \frac{sin ( x )}{cos ^{5} ( x )}

= \frac{sin ( x )}{cos ^{5} ( x )}

= \frac{sin ( x )}{cos ^{5} ( x )}

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

流行的例子

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