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受欢迎的三角函数 >

证明 2csc(2x)=sec(x)csc(x)

解答

证明

2 csc (2 x) = sec (x) csc (x)

解答

真

求解步骤

2 csc (2 x) = sec (x) csc (x)

调整左侧

2 csc (2 x)

用 sin, cos 表示

2 csc (2 x)

使用基本三角恒等式:

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

= 2 \cdot \frac{1}{sin ( 2 x )}

化简

2 \cdot \frac{1}{sin ( 2 x )} : \frac{2}{sin ( 2 x )}

2 \cdot \frac{1}{sin ( 2 x )}

分式相乘:

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

= \frac{1 \cdot 2}{sin ( 2 x )}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{sin ( 2 x )}

= \frac{2}{sin ( 2 x )}

= \frac{2}{sin ( 2 x )}

使用三角恒等式改写

\frac{2}{sin ( 2 x )}

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

= \frac{2}{2 sin ( x ) cos ( x )}

数字相除:

\frac{2}{2} = 1

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

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

使用基本三角恒等式:

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

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

化简

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

乘

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

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

分式相乘:

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

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

数字相乘:

1 \cdot 1 = 1

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

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

使用分式法则:

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

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

使用法则

\frac{a}{1} = a

= csc (x) sec (x)

csc (x) sec (x)

csc (x) sec (x)

= sec (x) csc (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 tan (\frac{x}{2}) = \frac{sin ( x )}{cos ( x ) + 1}

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p ro v e (1 - cos (θ)) (1 + sec (θ)) = sin (θ) tan (θ)

证明 tan(2t)=2sin(t)cos(t)sec(2t)

p ro v e tan (2 t) = 2 sin (t) cos (t) sec (2 t)

证明 tan^3(x)-tan(x)sec^2(x)=tan(-x)

p ro v e tan^{3} (x) - tan (x) sec^{2} (x) = tan (- x)

证明 sec(pi-x)=-sec(x)

p ro v e sec (π - x) = - sec (x)