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Beliebt Algebra >

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

Lösung

faktorisieren

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

Lösung

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

Schritte zur Lösung

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

Wende Exponentenregel an:

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

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

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

Schreibe

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

um:

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

Faktorisiere

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

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

Vereinfache

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

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

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