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受欢迎的三角函数 >

证明 (sec(x)csc(x))/2 =csc(2x)

解答

证明

\frac{sec ( x ) csc ( x )}{2} = csc (2 x)

解答

真

求解步骤

\frac{sec ( x ) csc ( x )}{2} = csc (2 x)

调整右侧

csc (2 x)

用 sin, cos 表示

csc (2 x)

使用基本三角恒等式:

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

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

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

使用三角恒等式改写

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

使用倍角公式:

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

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

= \frac{1}{2 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}{2 \cdot \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}{2 \cdot \frac{1}{c s c ( x )} \cdot \frac{1}{s e c ( x )}}

化简

\frac{1}{2 \cdot \frac{1}{c s c ( x )} \cdot \frac{1}{s e c ( x )}}

乘

2 \cdot \frac{1}{csc ( x )} \cdot \frac{1}{sec ( x )} : \frac{2}{csc ( x ) sec ( x )}

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

分式相乘:

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

= \frac{1 \cdot 1 \cdot 2}{csc ( x ) sec ( x )}

数字相乘:

1 \cdot 1 \cdot 2 = 2

= \frac{2}{csc ( x ) sec ( x )}

= \frac{1}{\frac{2}{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 )}{2}

\frac{csc ( x ) sec ( x )}{2}

\frac{csc ( x ) sec ( x )}{2}

= \frac{sec ( x ) csc ( x )}{2}

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-sec^2(x)=-tan^2(x)-1

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

证明 tan(4x)*cos(4x)=sin(4x)

p ro v e tan (4 x) \cdot cos (4 x) = sin (4 x)

证明 tan(θ)=csc(θ)sec(θ)-cot(θ)

p ro v e tan (θ) = csc (θ) sec (θ) - cot (θ)

证明 sec(x)-sec(x)sin(x^2)=cos(x^2)

p ro v e sec (x) - sec (x) sin (x^{2}) = cos (x^{2})

证明 (1+cot^2(x))= 1/(sin^2(x))

p ro v e (1 + cot^{2} (x)) = \frac{1}{sin ^{2} ( x )}