English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

sec^{4} (θ) - tan^{4} (θ)

Faktorisiere

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

sec^{4} (θ) - tan^{4} (θ)

Schreibe

sec^{4} (θ) - tan^{4} (θ)

um:

(sec^{2} (θ))^{2} - (tan^{2} (θ))^{2}

sec^{4} (θ) - tan^{4} (θ)

Wende Exponentenregel an:

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

sec^{4} (θ) = (sec^{2} (θ))^{2}

= (sec^{2} (θ))^{2} - tan^{4} (θ)

Wende Exponentenregel an:

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

tan^{4} (θ) = (tan^{2} (θ))^{2}

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

Faktorisiere

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

sec^{2} (θ) - tan^{2} (θ)

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

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

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

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

Vereinfache

tan^{2} (θ) + 1 + tan^{2} (θ) : 2 tan^{2} (θ) + 1

tan^{2} (θ) + 1 + tan^{2} (θ)

Fasse gleiche Terme zusammen

= tan^{2} (θ) + tan^{2} (θ) + 1

Addiere gleiche Elemente:

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

= 2 tan^{2} (θ) + 1

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

Multipliziere aus

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

a = sec (θ), b = tan (θ)

= sec^{2} (θ) - tan^{2} (θ)

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

Verwende die Pythagoreische Identität:

sec^{2} (θ) = tan^{2} (θ) + 1

sec^{2} (θ) - tan^{2} (θ) = 1

= 1 \cdot (2 tan^{2} (θ) + 1)

Vereinfache

= 2 tan^{2} (θ) + 1

= 2 tan^{2} (θ) + 1

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

\Rightarrow Wahr

p ro v e cos (- \frac{π}{4}) = sin (\frac{π}{4})

p ro v e sin (u) cos (u) = \frac{1}{2} sin (2 u)

p ro v e tan (2 x) cos (2 x) = sin (2 x)

p ro v e tan (x) = 2 sin (x)

p ro v e \frac{1 - sin ^{4} ( a )}{cos ^{2} ( a )} = 2 - cos^{2} (a)