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

解

証明する

2 - 2 tan (x) cot (2 x) = sec^{2} (x)

解

真

解答ステップ

2 - 2 tan (x) cot (2 x) = sec^{2} (x)

左側を操作する

2 - 2 tan (x) cot (2 x)

サイン, コサインで表わす

2 - 2 cot (2 x) tan (x)

基本的な三角関数の公式を使用する:

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

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

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

簡素化

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

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

乗じる

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

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

分数を乗じる:

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

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

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

元を分数に変換する:

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

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

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

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

三角関数の公式を使用して書き換える

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

2倍角の公式を使用:

cos (2 x) = 2 cos^{2} (x) - 1

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

2倍角の公式を使用:

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

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

簡素化

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

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

- 2 (2 cos^{2} (x) - 1) sin (x) + 2 cos (x) \cdot 2 sin (x) cos (x) = - 2 sin (x) (2 cos^{2} (x) - 1) + 4 cos^{2} (x) sin (x)

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

2 cos (x) \cdot 2 sin (x) cos (x) = 4 cos^{2} (x) sin (x)

2 cos (x) \cdot 2 sin (x) cos (x)

数を乗じる:

2 \cdot 2 = 4

= 4 cos (x) sin (x) cos (x)

指数の規則を適用する:

a^{b} \cdot a^{c} = a^{b + c}

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

= 4 sin (x) cos^{1 + 1} (x)

数を足す:

1 + 1 = 2

= 4 sin (x) cos^{2} (x)

= - 2 sin (x) (2 cos^{2} (x) - 1) + 4 cos^{2} (x) sin (x)

= \frac{- 2 sin ( x ) ( 2 cos ^{2} ( x ) - 1 ) + 4 cos ^{2} ( x ) sin ( x )}{2 cos ( x ) sin ( x ) cos ( x )}

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

cos (x) \cdot 2 sin (x) cos (x)

指数の規則を適用する:

a^{b} \cdot a^{c} = a^{b + c}

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

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

数を足す:

1 + 1 = 2

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

= \frac{- 2 sin ( x ) ( 2 cos ^{2} ( x ) - 1 ) + 4 cos ^{2} ( x ) sin ( x )}{2 cos ^{2} ( x ) sin ( x )}

因数

- 2 (2 cos^{2} (x) - 1) sin (x) + 4 sin (x) cos^{2} (x) : 2 sin (x)

- 2 (2 cos^{2} (x) - 1) sin (x) + 4 sin (x) cos^{2} (x)

書き換え

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

共通項をくくり出す

2 sin (x)

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

拡張

2 cos^{2} (x) - (2 cos^{2} (x) - 1) : 1

- (- 1 + cos^{2} (x) \cdot 2) + 2 cos^{2} (x)

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

- (- 1 + cos^{2} (x) \cdot 2) : 1 - cos^{2} (x) \cdot 2

- (- 1 + cos^{2} (x) \cdot 2)

括弧を分配する

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

マイナス・プラスの規則を適用する

- (- a) = a, - (a) = - a

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

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

類似した元を足す:

- 2 cos^{2} (x) + 2 cos^{2} (x) = 0

= 1

= 2 \cdot 1 \cdot sin (x)

改良

= 2 sin (x)

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

キャンセル

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

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

数を割る:

\frac{2}{2} = 1

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

共通因数を約分する:

sin (x)

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

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

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

= \frac{1}{cos ^{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:

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

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

簡素化

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

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

(\frac{1}{sec ( x )})^{2}

指数の規則を適用する:

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

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

規則を適用

1^{a} = 1

1^{2} = 1

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

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

分数の規則を適用する:

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

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

規則を適用

\frac{a}{1} = a

= sec^{2} (x)

sec^{2} (x)

sec^{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 sec^2(B)-csc^2(B)=(tan(B)-cot(B))/(sin(B)cos(B))

p ro v e sec^{2} (B) - csc^{2} (B) = \frac{tan ( B ) - cot ( B )}{sin ( B ) cos ( B )}

証明する 2+sec(a)cos(a)=3

p ro v e 2 + sec (a) cos (a) = 3

証明する 5cos^2(θ)+2sin^2(θ)=3cos^2(θ)+2

p ro v e 5 cos^{2} (θ) + 2 sin^{2} (θ) = 3 cos^{2} (θ) + 2

証明する (sec(a))/(tan(a)+cot(a))=sin(a)

p ro v e \frac{sec ( a )}{tan ( a ) + cot ( a )} = sin (a)

証明する sin((9pi)/2+x)=cos(x)

p ro v e sin (\frac{9 π}{2} + x) = cos (x)