English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

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

Faktorisiere

- 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)

Schreibe

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

um:

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

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

Wende Exponentenregel an:

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

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

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

Wende Exponentenregel an:

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}

Wende Formel zur Differenz von zwei Quadraten an:

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))

Faktorisiere

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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))

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die Pythagoreische Identität:

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)

Addiere gleiche Elemente:

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

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

Multipliziere aus

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

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

Wende Formel zur Differenz von zwei Quadraten an:

(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))

Verwende die Pythagoreische Identität:

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

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

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

Vereinfache

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

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

Entferne die Klammern:

(- a) = - a

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

Multipliziere:

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

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

Setze Klammern

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

Wende Minus-Plus Regeln an

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

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

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

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

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

Wir haben gezeigt, dass beide Seiten die gleiche Form annehmen können

\Rightarrow Wahr

p ro v e (tan (x) + cot (x)) tan (x) = sec^{2} (x)

p ro v e (cos (x) + sin (x))^{2} - 2 sin (x) cos (x) = 1

p ro v e 1 - cos^{2} (x) = \frac{tan ^{2} ( x )}{sec ^{2} ( x )}

p ro v e \frac{( cos ^{2} ( x ) )}{( 1 - sin ( x ) )} = 1 + sin (x)

p ro v e 8 csc^{2} (x) - 3 cot^{2} (x) = 3 + 5 csc^{2} (x)