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受欢迎的三角函数 >

证明 cot(x)=(csc(x))/(sec(x))

解答

证明

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

解答

真

求解步骤

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

调整左侧

cot (x)

用 sin, cos 表示

cot (x)

使用基本三角恒等式:

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

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

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

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{cos ( x )}{\frac{1}{c s c ( x )}}

使用基本三角恒等式:

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

\frac{\frac{1}{s e c ( x )}}{\frac{1}{c s c ( x )}}

化简

\frac{\frac{1}{s e c ( x )}}{\frac{1}{c s c ( x )}}

分式相除:

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

= \frac{1 \cdot csc ( x )}{sec ( x ) \cdot 1}

整理后得

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

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

\frac{csc ( x )}{sec ( 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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p ro v e \frac{1}{1 + sin ( x )} + \frac{1}{1 - sin ( x )} = 2 + 2 tan^{2} (x)

证明 sec(2x)=(sec^2(x))/(1-tan^2(x))

p ro v e sec (2 x) = \frac{sec ^{2} ( x )}{1 - tan ^{2} ( x )}

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p ro v e \frac{sin ( 3 x ) - sin ( 5 x )}{2 cos ( 4 x )} = - sin (x)

证明 tan(a)+cot(a)=2csc(2a)

p ro v e tan (a) + cot (a) = 2 csc (2 a)

证明 (sec(t)-1)/(tan(t))=(tan(t))/(sec(t)+1)

p ro v e \frac{sec ( t ) - 1}{tan ( t )} = \frac{tan ( t )}{sec ( t ) + 1}