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受欢迎的三角函数 >

证明 csc(θ)=(sec(θ))/(tan(θ))

解答

证明

csc (θ) = \frac{sec ( θ )}{tan ( θ )}

解答

真

求解步骤

csc (θ) = \frac{sec ( θ )}{tan ( θ )}

调整右侧

\frac{sec ( θ )}{tan ( θ )}

用 sin, cos 表示

\frac{sec ( θ )}{tan ( θ )}

使用基本三角恒等式:

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

= \frac{\frac{1}{c o s ( θ )}}{tan ( θ )}

使用基本三角恒等式:

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

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

化简

\frac{\frac{1}{c o s ( θ )}}{\frac{s i n ( θ )}{c o s ( θ )}} : \frac{1}{sin ( θ )}

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

分式相除:

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

= \frac{1 \cdot cos ( θ )}{cos ( θ ) sin ( θ )}

整理后得

= \frac{cos ( θ )}{cos ( θ ) sin ( θ )}

约分:

cos (θ)

= \frac{1}{sin ( θ )}

= \frac{1}{sin ( θ )}

= \frac{1}{sin ( θ )}

使用三角恒等式改写

使用基本三角恒等式:

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

\frac{1}{\frac{1}{c s c ( θ )}}

化简

\frac{1}{\frac{1}{c s c ( θ )}}

使用分式法则:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{csc ( θ )}{1}

使用法则

\frac{a}{1} = a

= csc (θ)

csc (θ)

csc (θ)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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