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Beliebt Trigonometrie >

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

Lösung

beweisen

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

Lösung

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die Doppelwinkelidentität:

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

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

Vereinfache

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

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

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

Wende Formel für perfekte quadratische Gleichungen an:

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

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

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

Vereinfache

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

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

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

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

Wende Exponentenregel an:

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

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

Multipliziere die Zahlen:

2 \cdot 2 = 4

= cos^{4} (x)

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

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

Wende Exponentenregel an:

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

= sin^{2 \cdot 2} (x)

Multipliziere die Zahlen:

2 \cdot 2 = 4

= sin^{4} (x)

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

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

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

Addiere gleiche Elemente:

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

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

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

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

Faktorisiere

cos^{4} (x) + sin^{4} (x) + 2 cos^{2} (x) sin^{2} (x) : (sin^{2} (x) + cos^{2} (x))^{2}

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

Zerlege die Ausdrücke in Gruppen

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

Schreibe um in die Form

a x^{2} + b x y + c y^{2}

Lass

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

sein

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

Definition

Faktoren von

1 : 1

1

Teiler (Faktoren)

Finde die Primfaktoren von

1 :

Addiere 1

1

Die Faktoren von

1

1

Für alle zwei Faktoren gilt

u * v = 1,

prüfe, ob

u + v = 2

Prüfe

u = 1, v = 1 : u * v = 1, u + v = 2 \Rightarrow

Wahr

u = 1, v = 1

Gruppiere

(a x^{2} + ux y^{2}) + (v x^{2} y + c y^{2})

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

Zurücksetzen

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

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

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

Klammere

sin^{2} (x)

aus

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

aus

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

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

Wende Exponentenregel an:

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

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

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

Klammere gleiche Terme aus

sin^{2} (x)

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

Klammere

cos^{2} (x)

aus

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

aus

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

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

Wende Exponentenregel an:

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

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

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

Klammere gleiche Terme aus

cos^{2} (x)

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

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

Klammere gleiche Terme aus

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

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

Fasse zusammen

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die Pythagoreische Identität:

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

= 1^{2}

Wende Regel an

1^{a} = 1

= 1

= 1

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

\Rightarrow Wahr

Beliebte Beispiele

beweisen cos(x)(sec(x)-1)=1-cos(x)

p ro v e cos (x) (sec (x) - 1) = 1 - cos (x)

beweisen csc^2(θ)(1-cos^2(θ))=-cos(pi/2)

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

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

p ro v e 4 sin^{2} (x) + 4 cos^{2} (x) = 4

beweisen cos(2x)=(2-sec^2(x))/(sec^2(x))

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

beweisen cot(y)=(sin(2y))/(1-cos(2y))

p ro v e cot (y) = \frac{sin ( 2 y )}{1 - cos ( 2 y )}