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受欢迎的三角函数 >

证明 (csc(-x))/(sec(-x))=-cot(x)

解答

证明

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

解答

真

求解步骤

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

调整左侧

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

使用负角恒等式:

csc (- x) = - csc (x)

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

使用负角恒等式:

sec (- x) = sec (x)

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

分式相除:

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

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

整理后得

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

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

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

使用基本三角恒等式:

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

= - cot (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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