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Populaire Trigonométrie >

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

Solution

prouver

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

Solution

vrai

étapes des solutions

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

En manipulant le côté gauche

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

Factoriser

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

1 - sin^{4} (x)

Factoriser le terme commun

- 1

= - (sin^{4} (x) - 1)

Factoriser

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

sin^{4} (x) - 1

Récrire

sin^{4} (x) - 1

comme

(sin^{2} (x))^{2} - 1^{2}

sin^{4} (x) - 1

Récrire

1

comme

1^{2}

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

Appliquer la règle de l'exposant:

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

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

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

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

Appliquer la formule de différence de deux carrés :

x^{2} - y^{2} = (x + y) (x - y)

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

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

Factoriser

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

sin^{2} (x) - 1

Récrire

1

comme

1^{2}

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

Appliquer la formule de différence de deux carrés :

x^{2} - y^{2} = (x + y) (x - y)

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

= (sin (x) + 1) (sin (x) - 1)

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

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

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

Récrire en utilisant des identités trigonométriques

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

Utiliser l'identité hyperbolique:

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

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

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

Simplifier

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

Développer

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

(sin (x) + 1) (sin (x) - 1)

Appliquer la formule de différence de deux carrés :

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

a = sin (x), b = 1

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

Appliquer la règle

1^{a} = 1

1^{2} = 1

= sin^{2} (x) - 1

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

Utiliser l'identité hyperbolique:

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

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

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

Simplifier

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

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

Appliquer la règle

- (- a) = a

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

Développer

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

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

Appliquer la loi de la distribution:

a (b - c) = ab - a c

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

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

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

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

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

Appliquer la règle de l'exposant:

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

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

= cos^{2 + 2} (x)

Additionner les nombres :

2 + 2 = 4

= cos^{4} (x)

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

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

Additionner les éléments similaires :

cos^{4} (x) - cos^{4} (x) = 0

= 2 cos^{2} (x)

= 2 cos^{2} (x)

= 2 cos^{2} (x)

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

\Rightarrow vrai

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