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

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

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

Faktorisiere

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

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

Schreibe

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

um:

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

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

Wende Exponentenregel an:

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

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

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

Wende Exponentenregel an:

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

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

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

Faktorisiere

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

= (sin (u) + cos (u)) (sin (u) - cos (u))

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Multipliziere aus

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

(sin (u) + cos (u)) (sin (u) - cos (u))

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

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

Verwende die Doppelwinkelidentität:

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

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

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

Vereinfache

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

Verwende die Doppelwinkelidentität:

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

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

Verwende die Pythagoreische Identität:

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

= - (1 - 2 sin^{2} (u)) \cdot 1

Vereinfache

- (1 - 2 sin^{2} (u)) \cdot 1 : - 1 + 2 sin^{2} (u)

- (1 - 2 sin^{2} (u)) \cdot 1

Multipliziere:

(1 - 2 sin^{2} (u)) \cdot 1 = (1 - 2 sin^{2} (u))

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

Setze Klammern

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

Wende Minus-Plus Regeln an

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

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

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

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

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

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

\Rightarrow Wahr

tan^{4} (x) - sec^{4} (x) = - 2 sec^{2} (x) + 1

(tan (x) + cot (x)) tan (x) = sec^{2} (x)

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

1 - cos^{2} (x) = \frac{tan ^{2} ( x )}{sec ^{2} ( x )}

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