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beweisen cos^4(A)-sin^4(A)=cos(2A)

Lösung

beweisen

cos^{4} (A) - sin^{4} (A) = cos (2 A)

Wahr

Schritte zur Lösung

cos^{4} (A) - sin^{4} (A) = cos (2 A)

Manipuliere die linke Seite

cos^{4} (A) - sin^{4} (A)

Faktorisiere

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

cos^{4} (A) - sin^{4} (A)

Schreibe

cos^{4} (A) - sin^{4} (A)

um:

(cos^{2} (A))^{2} - (sin^{2} (A))^{2}

cos^{4} (A) - sin^{4} (A)

Wende Exponentenregel an:

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

sin^{4} (A) = (sin^{2} (A))^{2}

= cos^{4} (A) - (sin^{2} (A))^{2}

Wende Exponentenregel an:

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

cos^{4} (A) = (cos^{2} (A))^{2}

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

Faktorisiere

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

cos^{2} (A) - sin^{2} (A)

Wende Formel zur Differenz von zwei Quadraten an:

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

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

= (cos (A) + sin (A)) (cos (A) - sin (A))

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die Pythagoreische Identität:

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

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

Vereinfache

= (cos (A) + sin (A)) (cos (A) - sin (A))

Multipliziere aus

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

(cos (A) + sin (A)) (cos (A) - sin (A))

Wende Formel zur Differenz von zwei Quadraten an:

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

a = cos (A), b = sin (A)

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

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

Verwende die Doppelwinkelidentität:

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

= cos (2 A)

= cos (2 A)

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

\Rightarrow Wahr

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

p ro v e tan (θ) + 1 = sec (θ)

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

p ro v e sin (12 x) = 2 sin (6 x) cos (6 x)

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