English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Trigonometrie >

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

Lösung

beweisen

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

Lösung

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

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

Faktorisiere

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

1 - sin^{4} (a)

Klammere gleiche Terme aus

- 1

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

Faktorisiere

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

sin^{4} (a) - 1

Schreibe

sin^{4} (a) - 1

um:

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

sin^{4} (a) - 1

Schreibe

1

um:

1^{2}

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

Wende Exponentenregel an:

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

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

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

Faktorisiere

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

sin^{2} (a) - 1

Schreibe

1

um:

1^{2}

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

= (sin (a) + 1) (sin (a) - 1)

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

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die Pythagoreische Identität:

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

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

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

Vereinfache

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

Multipliziere aus

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

(sin (a) + 1) (sin (a) - 1)

Wende Formel zur Differenz von zwei Quadraten an:

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

a = sin (a), b = 1

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

Wende Regel an

1^{a} = 1

1^{2} = 1

= sin^{2} (a) - 1

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

Verwende die Pythagoreische Identität:

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

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

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

Vereinfache

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

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

Wende Regel an

- (- a) = a

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

Multipliziere aus

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

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

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

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

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

cos^{2} (a) cos^{2} (a)

Wende Exponentenregel an:

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

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

= cos^{2 + 2} (a)

Addiere die Zahlen:

2 + 2 = 4

= cos^{4} (a)

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

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

Addiere gleiche Elemente:

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

= 2 cos^{2} (a)

= 2 cos^{2} (a)

= 2 cos^{2} (a)

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

\Rightarrow Wahr

Beliebte Beispiele

beweisen (sec(-x))/(tan(x)+cot(x))=sin(x)

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

beweisen sqrt(2)sin(x+45)=sin(x)+cos(x)

p ro v e 2 sin (x + 4 5^{\circ}) = sin (x) + cos (x)

beweisen tan^2(x)sin^2(x)=tan^2(x)+sin^2(x)

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

beweisen cos(-pi/6)=sin(-pi/6+pi/2)

p ro v e cos (- \frac{π}{6}) = sin (- \frac{π}{6} + \frac{π}{2})

beweisen 3sin(2x-15)=cos(2x-15)

p ro v e 3 sin (2 x - 15) = cos (2 x - 15)