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

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

sin^{4} (θ) - cos^{4} (θ)

Faktorisiere

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

- cos^{4} (θ) + sin^{4} (θ)

Schreibe

sin^{4} (θ) - cos^{4} (θ)

um:

(sin^{2} (θ))^{2} - (cos^{2} (θ))^{2}

sin^{4} (θ) - cos^{4} (θ)

Wende Exponentenregel an:

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

sin^{4} (θ) = (sin^{2} (θ))^{2}

= (sin^{2} (θ))^{2} - cos^{4} (θ)

Wende Exponentenregel an:

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

cos^{4} (θ) = (cos^{2} (θ))^{2}

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

Faktorisiere

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

sin^{2} (θ) - cos^{2} (θ)

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

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

Multipliziere aus

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

a = sin (θ), b = cos (θ)

= sin^{2} (θ) - cos^{2} (θ)

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

Verwende die Doppelwinkelidentität:

cos^{2} (θ) - sin^{2} (θ) = cos (2 θ)

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

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

Vereinfache

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

Verwende die Doppelwinkelidentität:

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

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

Verwende die Pythagoreische Identität:

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

= - (2 cos^{2} (θ) - 1) \cdot 1

Vereinfache

- (2 cos^{2} (θ) - 1) \cdot 1 : - 2 cos^{2} (θ) + 1

- (2 cos^{2} (θ) - 1) \cdot 1

Multipliziere:

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

= - (2 cos^{2} (θ) - 1)

Setze Klammern

= - (2 cos^{2} (θ)) - (- 1)

Wende Minus-Plus Regeln an

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

= - 2 cos^{2} (θ) + 1

= - 2 cos^{2} (θ) + 1

= - 2 cos^{2} (θ) + 1

= 1 - 2 cos^{2} (θ)

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

\Rightarrow Wahr

p ro v e \frac{1 + tan ( x )}{1 - tan ( x )} + \frac{1 + cot ( x )}{1 - cot ( x )} = 0

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

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

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

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