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beweisen tan^4(θ)-sec^4(θ)=1-2sec^2(θ)

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

tan^{4} (θ) - sec^{4} (θ)

Faktorisiere

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

- sec^{4} (θ) + tan^{4} (θ)

Schreibe

tan^{4} (θ) - sec^{4} (θ)

um:

(tan^{2} (θ))^{2} - (sec^{2} (θ))^{2}

tan^{4} (θ) - sec^{4} (θ)

Wende Exponentenregel an:

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

sec^{4} (θ) = (sec^{2} (θ))^{2}

= tan^{4} (θ) - (sec^{2} (θ))^{2}

Wende Exponentenregel an:

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

tan^{4} (θ) = (tan^{2} (θ))^{2}

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

Faktorisiere

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

tan^{2} (θ) - sec^{2} (θ)

Wende Formel zur Differenz von zwei Quadraten an:

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

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

= (tan (θ) + sec (θ)) (tan (θ) - sec (θ))

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die Pythagoreische Identität:

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

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

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

Addiere gleiche Elemente:

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

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

Multipliziere aus

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

(tan (θ) + sec (θ)) (tan (θ) - sec (θ))

Wende Formel zur Differenz von zwei Quadraten an:

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

a = tan (θ), b = sec (θ)

= tan^{2} (θ) - sec^{2} (θ)

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

Verwende die Pythagoreische Identität:

sec^{2} (θ) = tan^{2} (θ) + 1

sec^{2} (θ) - tan^{2} (θ) = 1

= (- 1 + 2 sec^{2} (θ)) (- 1)

Vereinfache

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

(- 1 + 2 sec^{2} (θ)) (- 1)

Entferne die Klammern:

(- a) = - a

= - (- 1 + 2 sec^{2} (θ)) \cdot 1

Multipliziere:

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

= - (2 sec^{2} (θ) - 1)

Setze Klammern

= - (- 1) - (2 sec^{2} (θ))

Wende Minus-Plus Regeln an

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

= 1 - 2 sec^{2} (θ)

= 1 - 2 sec^{2} (θ)

= 1 - 2 sec^{2} (θ)

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

\Rightarrow Wahr

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

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

p ro v e \frac{cos ^{2} ( t )}{cot ( t )} = cos (t) sin (t)

p ro v e 4 sec^{2} (θ) - 3 = 1 + 4 tan^{2} (θ)

p ro v e - sin (x) = sin (x)