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

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

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

Faktorisiere

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

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

Schreibe

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

um:

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

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

Wende Exponentenregel an:

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

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

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

Wende Exponentenregel an:

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

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

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

Faktorisiere

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die Pythagoreische Identität:

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

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

Vereinfache

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

Multipliziere aus

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

(cos (t) + sin (t)) (cos (t) - sin (t))

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

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

Verwende die Doppelwinkelidentität:

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

= cos (2 t)

Verwende die Doppelwinkelidentität:

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

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

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

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

\Rightarrow Wahr

p ro v e 1 + tan^{2} (- θ) = sec^{2} (θ)

p ro v e \frac{tan ( x ) + tan ( y )}{cot ( x ) + cot ( y )} = tan (x) tan (y)

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

p ro v e 1 - cos (2 x) = tan (x) sin (2 x)

p ro v e 1 = sec^{2} (2 x) - tan^{2} (2 x)