English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

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

Faktorisiere

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

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

Schreibe

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

um:

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

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

Wende Exponentenregel an:

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

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

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

Wende Exponentenregel an:

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

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

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

Faktorisiere

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

= (cos (x) + sin (x)) (cos (x) - sin (x))

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die Pythagoreische Identität:

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

= (cos (x) + sin (x)) (cos (x) - sin (x)) \cdot 1

Vereinfache

= (cos (x) + sin (x)) (cos (x) - sin (x))

Multipliziere aus

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

(cos (x) + sin (x)) (cos (x) - sin (x))

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

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

Verwende die Doppelwinkelidentität:

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

= cos (2 x)

= cos (2 x)

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

\Rightarrow Wahr

4 sin^{2} (x) - 3 = 0

sec (\frac{3 π}{2})

cot (2 π)

sin^{2} (0)

cos (\frac{π}{8})