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beweisen cos(3b)=cos(b)(cos^2(b)-3sin^2(b))

Lösung

beweisen

cos (3 b) = cos (b) (cos^{2} (b) - 3 sin^{2} (b))

Wahr

Schritte zur Lösung

cos (3 b) = cos (b) (cos^{2} (b) - 3 sin^{2} (b))

Manipuliere die linke Seite

cos (3 b)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

cos (3 b)

Verwende die folgenden Identitäten:

cos (3 x) = cos^{3} (x) - 3 sin^{2} (x) cos (x)

cos (3 x)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

cos (3 x)

Schreibe um

= cos (2 x + x)

Benutze die Identität der Winkelsumme:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (2 x) cos (x) - sin (2 x) sin (x)

Verwende die Doppelwinkelidentität:

sin (2 x) = 2 sin (x) cos (x)

= cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x)

Vereinfache

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

cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x)

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

2 sin (x) cos (x) sin (x)

Wende Exponentenregel an:

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

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

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

Addiere die Zahlen:

1 + 1 = 2

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

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

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

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

Verwende die Doppelwinkelidentität:

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

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

Multipliziere aus

(cos^{2} (x) - sin^{2} (x)) cos (x) - 2 sin^{2} (x) cos (x) : cos^{3} (x) - 3 sin^{2} (x) cos (x)

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

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

Multipliziere aus

cos (x) (cos^{2} (x) - sin^{2} (x)) : cos^{3} (x) - sin^{2} (x) cos (x)

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

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

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

cos^{2} (x) cos (x) = cos^{3} (x)

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

Wende Exponentenregel an:

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

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

= cos^{2 + 1} (x)

Addiere die Zahlen:

2 + 1 = 3

= cos^{3} (x)

= cos^{3} (x) - sin^{2} (x) cos (x)

= cos^{3} (x) - sin^{2} (x) cos (x) - 2 sin^{2} (x) cos (x)

Addiere gleiche Elemente:

- sin^{2} (x) cos (x) - 2 sin^{2} (x) cos (x) = - 3 sin^{2} (x) cos (x)

= cos^{3} (x) - 3 sin^{2} (x) cos (x)

= cos^{3} (x) - 3 sin^{2} (x) cos (x)

= cos^{3} (b) - 3 sin^{2} (b) cos (b)

= cos^{3} (b) - 3 sin^{2} (b) cos (b)

Manipuliere die rechte Seite

cos (b) (cos^{2} (b) - 3 sin^{2} (b))

Multipliziere aus

(cos^{2} (b) - 3 sin^{2} (b)) cos (b) : cos^{3} (b) - 3 sin^{2} (b) cos (b)

(cos^{2} (b) - 3 sin^{2} (b)) cos (b)

= cos (b) (cos^{2} (b) - 3 sin^{2} (b))

Wende das Distributivgesetz an:

a (b - c) = ab - a c

a = cos (b), b = cos^{2} (b), c = 3 sin^{2} (b)

= cos (b) cos^{2} (b) - cos (b) \cdot 3 sin^{2} (b)

= cos^{2} (b) cos (b) - 3 sin^{2} (b) cos (b)

cos^{2} (b) cos (b) = cos^{3} (b)

cos^{2} (b) cos (b)

Wende Exponentenregel an:

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

cos^{2} (b) cos (b) = cos^{2 + 1} (b)

= cos^{2 + 1} (b)

Addiere die Zahlen:

2 + 1 = 3

= cos^{3} (b)

= cos^{3} (b) - 3 sin^{2} (b) cos (b)

= cos^{3} (b) - 3 cos (b) sin^{2} (b)

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

\Rightarrow Wahr

p ro v e sec (b) - (tan (b)) (sin (b)) = cos (b)

p ro v e sec (- x) \cdot sin (x) = tan (x)

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

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

p ro v e cos (θ) - sin (θ) sin (2 θ) = cos (θ) cos (2 θ)