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 sec(2a)=(csc^2(a))/(cot^2(a)-1)

解

証明する

sec (2 a) = \frac{csc ^{2} ( a )}{cot ^{2} ( a ) - 1}

解

真

解答ステップ

sec (2 a) = \frac{csc ^{2} ( a )}{cot ^{2} ( a ) - 1}

右側を操作する

\frac{csc ^{2} ( a )}{cot ^{2} ( a ) - 1}

サイン, コサインで表わす

\frac{csc ^{2} ( a )}{- 1 + cot ^{2} ( a )}

基本的な三角関数の公式を使用する:

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

= \frac{( \frac{1}{s i n ( a )} ) ^{2}}{- 1 + cot ^{2} ( a )}

基本的な三角関数の公式を使用する:

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

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

簡素化

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

\frac{( \frac{1}{s i n ( a )} ) ^{2}}{- 1 + ( \frac{c o s ( a )}{s i n ( a )} ) ^{2}}

指数の規則を適用する:

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

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

(\frac{1}{sin ( a )})^{2} = \frac{1}{sin ^{2} ( a )}

(\frac{1}{sin ( a )})^{2}

指数の規則を適用する:

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

= \frac{1 ^{2}}{sin ^{2} ( a )}

規則を適用

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( a )}

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

分数の規則を適用する:

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

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

結合

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

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

元を分数に変換する:

1 = \frac{1 sin ^{2} ( a )}{sin ^{2} ( a )}

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

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 sin ^{2} ( a ) + cos ^{2} ( a )}{sin ^{2} ( a )}

乗算:

1 \cdot sin^{2} (a) = sin^{2} (a)

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

= \frac{1}{\frac{c o s ^{2} ( a ) - s i n ^{2} ( a )}{s i n ^{2} ( a )} sin ^{2} ( a )}

乗じる

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

sin^{2} (a) \frac{- sin ^{2} ( a ) + cos ^{2} ( a )}{sin ^{2} ( a )}

分数を乗じる:

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

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

共通因数を約分する:

sin^{2} (a)

= - - sin^{2} (a) + cos^{2} (a)

= \frac{1}{- sin ^{2} ( a ) + cos ^{2} ( a )}

= \frac{1}{- sin ^{2} ( a ) + cos ^{2} ( a )}

= \frac{1}{cos ^{2} ( a ) - sin ^{2} ( a )}

三角関数の公式を使用して書き換える

\frac{1}{cos ^{2} ( a ) - sin ^{2} ( a )}

2倍角の公式を使用:

cos^{2} (x) - sin^{2} (x) = cos (2 x)

= \frac{1}{cos ( 2 a )}

= \frac{1}{cos ( 2 a )}

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:

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

\frac{1}{\frac{1}{s e c ( 2 a )}}

簡素化

\frac{1}{\frac{1}{s e c ( 2 a )}}

分数の規則を適用する:

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

= \frac{sec ( 2 a )}{1}

規則を適用

\frac{a}{1} = a

= sec (2 a)

sec (2 a)

sec (2 a)

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(2α))/(1+cos(2α))=tan(α)

p ro v e \frac{sin ( 2 α )}{1 + cos ( 2 α )} = tan (α)

証明する cos(t)+tan(t)sin(t)=sec(t)

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

証明する cos(4θ)=1-8sin^2(θ)cos^2(θ)

p ro v e cos (4 θ) = 1 - 8 sin^{2} (θ) cos^{2} (θ)

証明する cos^6(105)=((1+cos(210))/2)^3

p ro v e cos^{6} (10 5^{\circ}) = (\frac{1 + cos ( 21 0 ^{\circ} )}{2})^{3}

証明する 1+tan^2(45)=sec^2(45)

p ro v e 1 + tan^{2} (4 5^{\circ}) = sec^{2} (4 5^{\circ})