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prouver tan^4(x)-sec^4(x)=-2sec^2(x)+1

Solution

prove

vrai

étapes des solutions

tan^{4} (x) - sec^{4} (x) = - 2 sec^{2} (x) + 1

En manipulant le côté gauche

tan^{4} (x) - sec^{4} (x)

Factoriser

- sec^{4} (x) + tan^{4} (x) : (tan^{2} (x) + sec^{2} (x)) (tan (x) + sec (x)) (tan (x) - sec (x))

- sec^{4} (x) + tan^{4} (x)

Récrire

tan^{4} (x) - sec^{4} (x)

comme

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

tan^{4} (x) - sec^{4} (x)

Appliquer la règle de l'exposant:

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

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

= tan^{4} (x) - (sec^{2} (x))^{2}

Appliquer la règle de l'exposant:

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

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

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

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

Appliquer la formule de différence de deux carrés :

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

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

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

Factoriser

tan^{2} (x) - sec^{2} (x) : (tan (x) + sec (x)) (tan (x) - sec (x))

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

Appliquer la formule de différence de deux carrés :

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

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

= (tan (x) + sec (x)) (tan (x) - sec (x))

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

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

Récrire en utilisant des identités trigonométriques

(- sec (x) + tan (x)) (sec (x) + tan (x)) (sec^{2} (x) + tan^{2} (x))

Utiliser l'identité hyperbolique:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= (- sec (x) + tan (x)) (sec (x) + tan (x)) (sec^{2} (x) + sec^{2} (x) - 1)

Additionner les éléments similaires :

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

= (tan (x) - sec (x)) (sec (x) + tan (x)) (2 sec^{2} (x) - 1)

Développer

(tan (x) + sec (x)) (tan (x) - sec (x)) : tan^{2} (x) - sec^{2} (x)

(tan (x) + sec (x)) (tan (x) - sec (x))

Appliquer la formule de différence de deux carrés :

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

a = tan (x), b = sec (x)

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

= (- 1 + 2 sec^{2} (x)) (tan^{2} (x) - sec^{2} (x))

Utiliser l'identité hyperbolique:

sec^{2} (x) = tan^{2} (x) + 1

sec^{2} (x) - tan^{2} (x) = 1

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

Simplifier

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

(- 1 + 2 sec^{2} (x)) (- 1)

Retirer les parenthèses:

(- a) = - a

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

Multiplier:

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

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

Distribuer des parenthèses

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

Appliquer les règles des moins et des plus

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

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

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

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

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

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

\Rightarrow vrai