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

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

cos^{4} (β) - sin^{4} (β)

Faktorisiere

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

cos^{4} (β) - sin^{4} (β)

Schreibe

cos^{4} (β) - sin^{4} (β)

um:

(cos^{2} (β))^{2} - (sin^{2} (β))^{2}

cos^{4} (β) - sin^{4} (β)

Wende Exponentenregel an:

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

sin^{4} (β) = (sin^{2} (β))^{2}

= cos^{4} (β) - (sin^{2} (β))^{2}

Wende Exponentenregel an:

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

cos^{4} (β) = (cos^{2} (β))^{2}

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

Faktorisiere

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

cos^{2} (β) - sin^{2} (β)

Wende Formel zur Differenz von zwei Quadraten an:

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

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

= (cos (β) + sin (β)) (cos (β) - sin (β))

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die Pythagoreische Identität:

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

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

Vereinfache

= (cos (β) + sin (β)) (cos (β) - sin (β))

Multipliziere aus

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

(cos (β) + sin (β)) (cos (β) - sin (β))

Wende Formel zur Differenz von zwei Quadraten an:

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

a = cos (β), b = sin (β)

= cos^{2} (β) - sin^{2} (β)

= cos^{2} (β) - sin^{2} (β)

Verwende die Doppelwinkelidentität:

cos^{2} (β) - sin^{2} (β) = cos (2 β)

= cos (2 β)

Verwende die Doppelwinkelidentität:

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

= 2 cos^{2} (β) - 1

= 2 cos^{2} (β) - 1

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

\Rightarrow Wahr

p ro v e (1 - sin^{2} (x)) \cdot csc^{2} (x) = cot^{2} (x)

p ro v e cot^{2} (θ) - cos^{2} (θ) = cot^{2} (θ) cos^{2} (θ)

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

p ro v e sin (x) cos (x) + sin^{3} (x) sec (x) = tan (x)

p ro v e - tan (θ) cos (θ) = sin (- θ)