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受欢迎的三角函数 >

证明 (8csc(-x))/(sec(-x))=-8cot(x)

解答

证明

\frac{8 csc ( - x )}{sec ( - x )} = - 8 cot (x)

解答

真

求解步骤

\frac{8 csc ( - x )}{sec ( - x )} = - 8 cot (x)

调整左侧

\frac{8 csc ( - x )}{sec ( - x )}

使用负角恒等式:

csc (- x) = - csc (x)

= \frac{8 ( - csc ( x ) )}{sec ( - x )}

使用负角恒等式:

sec (- x) = sec (x)

= \frac{8 ( - csc ( x ) )}{sec ( x )}

用 sin, cos 表示

\frac{( - csc ( x ) ) \cdot 8}{sec ( x )}

使用基本三角恒等式:

csc (x) = \frac{1}{sin ( x )}

= \frac{( - \frac{1}{s i n ( x )} ) \cdot 8}{sec ( x )}

使用基本三角恒等式:

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

= \frac{( - \frac{1}{s i n ( x )} ) \cdot 8}{\frac{1}{c o s ( x )}}

化简

\frac{( - \frac{1}{s i n ( x )} ) \cdot 8}{\frac{1}{c o s ( x )}} : - \frac{8 cos ( x )}{sin ( x )}

\frac{( - \frac{1}{s i n ( x )} ) \cdot 8}{\frac{1}{c o s ( x )}}

去除括号:

(- a) = - a

= \frac{- \frac{1}{s i n ( x )} \cdot 8}{\frac{1}{c o s ( x )}}

使用分式法则:

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

= - \frac{\frac{1}{s i n ( x )} \cdot 8}{\frac{1}{c o s ( x )}}

使用分式法则:

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

\frac{\frac{1}{s i n ( x )} \cdot 8}{\frac{1}{c o s ( x )}} = \frac{\frac{1}{s i n ( x )} \cdot 8 cos ( x )}{1}

= - \frac{\frac{1}{s i n ( x )} \cdot 8 cos ( x )}{1}

使用分式法则:

\frac{a}{1} = a

\frac{\frac{1}{s i n ( x )} \cdot 8 cos ( x )}{1} = \frac{1}{sin ( x )} \cdot 8 cos (x)

= - \frac{1}{sin ( x )} \cdot 8 cos (x)

分式相乘:

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

= - \frac{1 \cdot 8 cos ( x )}{sin ( x )}

数字相乘:

1 \cdot 8 = 8

= - \frac{8 cos ( x )}{sin ( x )}

= - \frac{8 cos ( x )}{sin ( x )}

= - \frac{8 cos ( x )}{sin ( x )}

使用三角恒等式改写

= - 8 \cdot \frac{cos ( x )}{sin ( x )}

使用基本三角恒等式:

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

- 8 cot (x)

- 8 cot (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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