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 cos^2(2x)+4sin^2(x)cos^2(x)=1

解

証明する

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

解

真

解答ステップ

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

左側を操作する

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

三角関数の公式を使用して書き換える

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

2倍角の公式を使用:

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

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

簡素化

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

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

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

完全平方式を適用する:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = cos^{2} (x), b = sin^{2} (x)

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

簡素化

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

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

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

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

指数の規則を適用する:

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

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

数を乗じる:

2 \cdot 2 = 4

= cos^{4} (x)

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

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

指数の規則を適用する:

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

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

数を乗じる:

2 \cdot 2 = 4

= sin^{4} (x)

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

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

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

類似した元を足す:

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

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

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

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

因数

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

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

式をグループに分ける

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

a x^{2} + b x y + c y^{2}

形式で書き直す

仮に

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

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

定義

以下の因数:

1 : 1

1

除数 (因数)

以下の素因数を求める:

1 :

1 を加える

1

以下の因数:

1

1

u * v = 1

などの各 2 因数で以下をチェックする:

u + v = 2

以下をチェックする:

u = 1, v = 1 : u * v = 1, u + v = 2 \Rightarrow

真

u = 1, v = 1

以下に分ける:

(a x^{2} + ux y^{2}) + (v x^{2} y + c y^{2})

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

リセット

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

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

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

sin^{2} (x)

を

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

からくくり出す

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

指数の規則を適用する:

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

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

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

共通項をくくり出す

sin^{2} (x)

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

cos^{2} (x)

を

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

からくくり出す

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

指数の規則を適用する:

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

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

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

共通項をくくり出す

cos^{2} (x)

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

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

共通項をくくり出す

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

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

改良

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

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

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

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

= 1^{2}

規則を適用

1^{a} = 1

= 1

= 1

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 cos(x)(sec(x)-1)=1-cos(x)

p ro v e cos (x) (sec (x) - 1) = 1 - cos (x)

証明する csc^2(θ)(1-cos^2(θ))=-cos(pi/2)

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

証明する 4sin^2(x)+4cos^2(x)=4

p ro v e 4 sin^{2} (x) + 4 cos^{2} (x) = 4

証明する cos(2x)=(2-sec^2(x))/(sec^2(x))

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

証明する cot(y)=(sin(2y))/(1-cos(2y))

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