English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

sin^{4} (y) - cos^{4} (y)

Faktorisiere

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

- cos^{4} (y) + sin^{4} (y)

Schreibe

sin^{4} (y) - cos^{4} (y)

um:

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

sin^{4} (y) - cos^{4} (y)

Wende Exponentenregel an:

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

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

= (sin^{2} (y))^{2} - cos^{4} (y)

Wende Exponentenregel an:

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

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

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

Faktorisiere

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

= (sin (y) + cos (y)) (sin (y) - cos (y))

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Multipliziere aus

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

(sin (y) + cos (y)) (sin (y) - cos (y))

Wende Formel zur Differenz von zwei Quadraten an:

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

a = sin (y), b = cos (y)

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

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

Verwende die Doppelwinkelidentität:

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

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

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

Vereinfache

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

Verwende die Doppelwinkelidentität:

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

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

Verwende die Pythagoreische Identität:

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

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

Vereinfache

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

- (2 cos^{2} (y) - 1) \cdot 1

Multipliziere:

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

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

Setze Klammern

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

Wende Minus-Plus Regeln an

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

= - 2 cos^{2} (y) + 1

= - 2 cos^{2} (y) + 1

= - 2 cos^{2} (y) + 1

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

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

\Rightarrow Wahr

p ro v e 1 - sin^{2} (- θ) = cos^{2} (θ)

p ro v e tan (θ) csc^{2} (θ) - tan (θ) = cot (θ)

p ro v e cos (α + β) + cos (α - β) = 2 cos (α) cos (β)

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

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