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受欢迎的三角函数 >

证明 cot(a)sec(a)=csc(a)

解答

证明

cot (a) sec (a) = csc (a)

解答

真

求解步骤

cot (a) sec (a) = csc (a)

调整左侧

cot (a) sec (a)

用 sin, cos 表示

cot (a) sec (a)

使用基本三角恒等式:

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

= \frac{cos ( a )}{sin ( a )} sec (a)

使用基本三角恒等式:

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

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

化简

\frac{cos ( a )}{sin ( a )} \cdot \frac{1}{cos ( a )} : \frac{1}{sin ( a )}

\frac{cos ( a )}{sin ( a )} \cdot \frac{1}{cos ( a )}

分式相乘:

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

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

约分:

cos (a)

= \frac{1}{sin ( a )}

= \frac{1}{sin ( a )}

= \frac{1}{sin ( a )}

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 ( a )}}

化简

\frac{1}{\frac{1}{c s c ( a )}}

使用分式法则:

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

= \frac{csc ( a )}{1}

使用法则

\frac{a}{1} = a

= csc (a)

csc (a)

csc (a)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

流行的例子

证明 tan(x/2)+cot(x/2)=2csc(x)

p ro v e tan (\frac{x}{2}) + cot (\frac{x}{2}) = 2 csc (x)

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

证明 sec(u)+tan(u)=(cos(u))/(1-sin(u))

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

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p ro v e cos (x - \frac{π}{6}) - cos (x + \frac{π}{6}) = sin (x)

证明 tan(2θ)=(2tan(θ))/(1-tan^2(θ))

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