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beweisen cos^4(θ)-sin^4(θ)=2cos^2(θ)-1

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

cos^{4} (θ) - sin^{4} (θ)

Faktorisiere

cos^{4} (θ) - sin^{4} (θ) : (cos^{2} (θ) + sin^{2} (θ)) (cos (θ) + sin (θ)) (cos (θ) - sin (θ))

cos^{4} (θ) - sin^{4} (θ)

Schreibe

cos^{4} (θ) - sin^{4} (θ)

um:

(cos^{2} (θ))^{2} - (sin^{2} (θ))^{2}

cos^{4} (θ) - sin^{4} (θ)

Wende Exponentenregel an:

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

sin^{4} (θ) = (sin^{2} (θ))^{2}

= cos^{4} (θ) - (sin^{2} (θ))^{2}

Wende Exponentenregel an:

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

cos^{4} (θ) = (cos^{2} (θ))^{2}

= (cos^{2} (θ))^{2} - (sin^{2} (θ))^{2}

= (cos^{2} (θ))^{2} - (sin^{2} (θ))^{2}

Wende Formel zur Differenz von zwei Quadraten an:

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

(cos^{2} (θ))^{2} - (sin^{2} (θ))^{2} = (cos^{2} (θ) + sin^{2} (θ)) (cos^{2} (θ) - sin^{2} (θ))

= (cos^{2} (θ) + sin^{2} (θ)) (cos^{2} (θ) - sin^{2} (θ))

Faktorisiere

cos^{2} (θ) - sin^{2} (θ) : (cos (θ) + sin (θ)) (cos (θ) - sin (θ))

cos^{2} (θ) - sin^{2} (θ)

Wende Formel zur Differenz von zwei Quadraten an:

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

cos^{2} (θ) - sin^{2} (θ) = (cos (θ) + sin (θ)) (cos (θ) - sin (θ))

= (cos (θ) + sin (θ)) (cos (θ) - sin (θ))

= (cos^{2} (θ) + sin^{2} (θ)) (cos (θ) + sin (θ)) (cos (θ) - sin (θ))

= (cos (θ) + sin (θ)) (cos (θ) - sin (θ)) (cos^{2} (θ) + sin^{2} (θ))

Umschreiben mit Hilfe von Trigonometrie-Identitäten

(cos (θ) + sin (θ)) (cos (θ) - sin (θ)) (cos^{2} (θ) + sin^{2} (θ))

Verwende die Pythagoreische Identität:

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

= (cos (θ) + sin (θ)) (cos (θ) - sin (θ)) \cdot 1

Vereinfache

= (cos (θ) + sin (θ)) (cos (θ) - sin (θ))

Multipliziere aus

(cos (θ) + sin (θ)) (cos (θ) - sin (θ)) : cos^{2} (θ) - sin^{2} (θ)

(cos (θ) + sin (θ)) (cos (θ) - sin (θ))

Wende Formel zur Differenz von zwei Quadraten an:

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

a = cos (θ), b = sin (θ)

= cos^{2} (θ) - sin^{2} (θ)

= cos^{2} (θ) - sin^{2} (θ)

Verwende die Doppelwinkelidentität:

cos^{2} (θ) - sin^{2} (θ) = cos (2 θ)

= cos (2 θ)

Verwende die Doppelwinkelidentität:

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

= 2 cos^{2} (θ) - 1

= 2 cos^{2} (θ) - 1

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

\Rightarrow Wahr

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

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

p ro v e cos (4 θ) = 8 cos^{4} (θ) - 8 cos^{2} (θ) + 1

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

p ro v e sin (2 A) = 2 sin (A) cos (A)