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受欢迎的三角函数 >

证明 (csc(x))/(2cos(x))=csc(2x)

解答

证明

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

解答

真

求解步骤

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

化简

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

乘

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

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

分式相乘:

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

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

数字相乘:

1 \cdot 2 = 2

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

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

使用分式法则:

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

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

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

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

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^2(u)=(1-cos(2u))/(1+cos(2u))

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

证明 csc^4(x)-csc^2(x)=cot^2(+cot^4(x))

p ro v e csc^{4} (x) - csc^{2} (x) = cot^{2} (+ cot^{4} (x))

证明 (cot^2(x))/(csc(x)+1)+1=csc(x)

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

证明 (sin^2(θ))/(1-cos(θ))-1=cos(θ)

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

证明 1-sin(y)=(cos^2(y))/(1+sin(y))

p ro v e 1 - sin (y) = \frac{cos ^{2} ( y )}{1 + sin ( y )}