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Beliebt Trigonometrie >

beweisen 1+cos^4(x)-sin^4(x)=2cos^2(x)

Lösung

beweisen

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

Lösung

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

1 + cos^{4} (x) - sin^{4} (x)

Faktorisiere

1 - sin^{4} (x) : - (sin^{2} (x) + 1) (sin (x) + 1) (sin (x) - 1)

1 - sin^{4} (x)

Klammere gleiche Terme aus

- 1

= - (sin^{4} (x) - 1)

Faktorisiere

sin^{4} (x) - 1 : (sin^{2} (x) + 1) (sin (x) + 1) (sin (x) - 1)

sin^{4} (x) - 1

Schreibe

sin^{4} (x) - 1

um:

(sin^{2} (x))^{2} - 1^{2}

sin^{4} (x) - 1

Schreibe

1

um:

1^{2}

= sin^{4} (x) - 1^{2}

Wende Exponentenregel an:

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

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

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

Faktorisiere

sin^{2} (x) - 1 : (sin (x) + 1) (sin (x) - 1)

sin^{2} (x) - 1

Schreibe

1

um:

1^{2}

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

= (sin (x) + 1) (sin (x) - 1)

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

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

cos^{4} (x) - (- 1 + sin (x)) (1 + sin (x)) (1 + sin^{2} (x))

Verwende die Pythagoreische Identität:

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

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

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

Vereinfache

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

Multipliziere aus

(sin (x) + 1) (sin (x) - 1) : sin^{2} (x) - 1

(sin (x) + 1) (sin (x) - 1)

Wende Formel zur Differenz von zwei Quadraten an:

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

a = sin (x), b = 1

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

Wende Regel an

1^{a} = 1

1^{2} = 1

= sin^{2} (x) - 1

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

Verwende die Pythagoreische Identität:

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

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

= cos^{4} (x) - (- cos^{2} (x)) (2 - cos^{2} (x))

Vereinfache

cos^{4} (x) - (- cos^{2} (x)) (2 - cos^{2} (x)) : 2 cos^{2} (x)

cos^{4} (x) - (- cos^{2} (x)) (2 - cos^{2} (x))

Wende Regel an

- (- a) = a

= cos^{4} (x) + cos^{2} (x) (2 - cos^{2} (x))

Multipliziere aus

cos^{2} (x) (2 - cos^{2} (x)) : 2 cos^{2} (x) - cos^{4} (x)

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

a = cos^{2} (x), b = 2, c = cos^{2} (x)

= cos^{2} (x) \cdot 2 - cos^{2} (x) cos^{2} (x)

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

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

cos^{2} (x) cos^{2} (x)

Wende Exponentenregel an:

a^{b} \cdot a^{c} = a^{b + c}

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

= cos^{2 + 2} (x)

Addiere die Zahlen:

2 + 2 = 4

= cos^{4} (x)

= 2 cos^{2} (x) - cos^{4} (x)

= cos^{4} (x) + 2 cos^{2} (x) - cos^{4} (x)

Addiere gleiche Elemente:

cos^{4} (x) - cos^{4} (x) = 0

= 2 cos^{2} (x)

= 2 cos^{2} (x)

= 2 cos^{2} (x)

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

\Rightarrow Wahr

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