English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Populaire Trigonométrie >

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

Solution

prouver

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

Solution

vrai

étapes des solutions

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

En manipulant le côté gauche

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

Récrire en utilisant des identités trigonométriques

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

Utiliser l'identité d'angle double:

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

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

Simplifier

(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)

Appliquer la formule du carré parfait:

(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}

Simplifier

(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}

Appliquer la règle de l'exposant:

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

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

Multiplier les nombres :

2 \cdot 2 = 4

= cos^{4} (x)

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

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

Appliquer la règle de l'exposant:

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

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

Multiplier les nombres :

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)

Additionner les éléments similaires :

- 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)

Factoriser

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)

Décomposer l'expression en groupes

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

Récrire sous la forme

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

Soit

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

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

Définition

Facteurs de

1 : 1

1

Diviseurs (Facteurs)

Trouver les facteurs premiers de

1 :

Ajouter 1

1

Les facteurs de

1

1

Pour chaque deux facteurs tels que

u * v = 1,

vérifier si

u + v = 2

Vérifier

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

vrai

u = 1, v = 1

Grouper dans

(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))

Redéfinir

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))

Factoriser

sin^{2} (x)

depuis

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

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

Appliquer la règle de l'exposant:

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)

Factoriser le terme commun

sin^{2} (x)

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

Factoriser

cos^{2} (x)

depuis

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

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

Appliquer la règle de l'exposant:

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)

Factoriser le terme commun

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))

Factoriser le terme commun

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

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

Redéfinir

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

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

Récrire en utilisant des identités trigonométriques

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

Utiliser l'identité hyperbolique:

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

= 1^{2}

Appliquer la règle

1^{a} = 1

= 1

= 1

Nous avons démontré que les deux côtés pourraient avoir la même forme

\Rightarrow vrai

Exemples populaires

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

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

prouver 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})

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

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

prouver cos(2x)=(2-sec^2(x))/(sec^2(x))

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

prouver cot(y)=(sin(2y))/(1-cos(2y))

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