ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある三角関数 >

証明する csc(2x)=(1+tan^2(x))/(2tan(x))

解

証明する

csc (2 x) = \frac{1 + tan ^{2} ( x )}{2 tan ( x )}

解

真

解答ステップ

csc (2 x) = \frac{1 + tan ^{2} ( x )}{2 tan ( x )}

右側を操作する

\frac{1 + tan ^{2} ( x )}{2 tan ( x )}

サイン, コサインで表わす

\frac{1 + tan ^{2} ( x )}{2 tan ( x )}

基本的な三角関数の公式を使用する:

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

= \frac{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{2 \cdot \frac{s i n ( x )}{c o s ( x )}}

簡素化

\frac{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{2 \cdot \frac{s i n ( x )}{c o s ( x )}} : \frac{cos ^{2} ( x ) + sin ^{2} ( x )}{2 cos ( x ) sin ( x )}

\frac{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{2 \cdot \frac{s i n ( x )}{c o s ( x )}}

乗じる

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

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

分数を乗じる:

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

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

= \frac{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{\frac{2 s i n ( x )}{c o s ( x )}}

指数の規則を適用する:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 + \frac{s i n ^{2} ( x )}{c o s ^{2} ( x )}}{\frac{2 s i n ( x )}{c o s ( x )}}

分数の規則を適用する:

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

= \frac{( 1 + \frac{s i n ^{2} ( x )}{c o s ^{2} ( x )} ) cos ( x )}{sin ( x ) \cdot 2}

結合

1 + \frac{sin ^{2} ( x )}{cos ^{2} ( x )} : \frac{cos ^{2} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x )}

1 + \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

元を分数に変換する:

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

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

分母が等しいので, 分数を組み合わせる:

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

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

乗算:

1 \cdot cos^{2} (x) = cos^{2} (x)

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

= \frac{\frac{c o s ^{2} ( x ) + s i n ^{2} ( x )}{c o s ^{2} ( x )} cos ( x )}{2 sin ( x )}

乗じる

\frac{cos ^{2} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x )} cos (x) : \frac{cos ^{2} ( x ) + sin ^{2} ( x )}{cos ( x )}

\frac{cos ^{2} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x )} cos (x)

分数を乗じる:

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

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

共通因数を約分する:

cos (x)

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

= \frac{\frac{c o s ^{2} ( x ) + s i n ^{2} ( x )}{c o s ( x )}}{2 sin ( x )}

分数の規則を適用する:

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

= \frac{cos ^{2} ( x ) + sin ^{2} ( x )}{cos ( x ) sin ( x ) \cdot 2}

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

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

三角関数の公式を使用して書き換える

\frac{cos ^{2} ( x ) + sin ^{2} ( x )}{2 cos ( x ) sin ( x )}

2倍角の公式を使用:

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

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

ピタゴラスの公式を使用する:

cos^{2} (x) + sin^{2} (x) = 1

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

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

三角関数の公式を使用して書き換える

基本的な三角関数の公式を使用する:

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

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

簡素化

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

分数の規則を適用する:

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

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

規則を適用

\frac{a}{1} = a

= csc (2 x)

csc (2 x)

csc (2 x)

両辺を同じ形式にできることを証明した

\Rightarrow 真

人気の例

証明する (sin(x))/(1-cos(x))=cot(x)+csc(x)

p ro v e \frac{sin ( x )}{1 - cos ( x )} = cot (x) + csc (x)

証明する tan(30)=(sin(30))/(cos(30))

p ro v e tan (3 0^{\circ}) = \frac{sin ( 3 0 ^{\circ} )}{cos ( 3 0 ^{\circ} )}

証明する tan(2x)-sin(2x)=tan(2x)sin(2x)

p ro v e tan (2 x) - sin (2 x) = tan (2 x) sin (2 x)

証明する (2-sec^2(x))/(sec^2(x))=cos(2x)

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

証明する (cos(β))/(1-sin(β))=sec(β)+tan(β)

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