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Beliebt Trigonometrie >

beweisen (1-sin^4(a))/(cos^2(a))=2-cos^2(a)

Lösung

beweisen

\frac{1 - sin ^{4} ( a )}{cos ^{2} ( a )} = 2 - cos^{2} (a)

Lösung

Wahr

Schritte zur Lösung

\frac{1 - sin ^{4} ( a )}{cos ^{2} ( a )} = 2 - cos^{2} (a)

Manipuliere die linke Seite

\frac{1 - sin ^{4} ( a )}{cos ^{2} ( a )}

Umschreiben mit Hilfe von Trigonometrie-Identitäten

\frac{1 - sin ^{4} ( a )}{cos ^{2} ( a )}

Verwende die Pythagoreische Identität:

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

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

= \frac{1 - sin ^{4} ( a )}{1 - sin ^{2} ( a )}

Vereinfache

\frac{1 - sin ^{4} ( a )}{1 - sin ^{2} ( a )} : sin^{2} (a) + 1

\frac{1 - sin ^{4} ( a )}{1 - sin ^{2} ( a )}

Faktorisiere

1 - sin^{4} (a) : - (sin^{2} (a) + 1) (sin (a) + 1) (sin (a) - 1)

1 - sin^{4} (a)

Klammere gleiche Terme aus

- 1

= - (sin^{4} (a) - 1)

Faktorisiere

sin^{4} (a) - 1 : (sin^{2} (a) + 1) (sin (a) + 1) (sin (a) - 1)

sin^{4} (a) - 1

Schreibe

sin^{4} (a) - 1

um:

(sin^{2} (a))^{2} - 1^{2}

sin^{4} (a) - 1

Schreibe

1

um:

1^{2}

= sin^{4} (a) - 1^{2}

Wende Exponentenregel an:

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

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

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

Faktorisiere

sin^{2} (a) - 1 : (sin (a) + 1) (sin (a) - 1)

sin^{2} (a) - 1

Schreibe

1

um:

1^{2}

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

= (sin (a) + 1) (sin (a) - 1)

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

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

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

Faktorisiere

1 - sin^{2} (a) : - (sin (a) + 1) (sin (a) - 1)

1 - sin^{2} (a)

Klammere gleiche Terme aus

- 1

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

Faktorisiere

sin^{2} (a) - 1 : (sin (a) + 1) (sin (a) - 1)

sin^{2} (a) - 1

Schreibe

1

um:

1^{2}

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

= (sin (a) + 1) (sin (a) - 1)

= - (sin (a) + 1) (sin (a) - 1)

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

Streiche

\frac{( sin ^{2} ( a ) + 1 ) ( sin ( a ) + 1 ) ( sin ( a ) - 1 )}{( sin ( a ) + 1 ) ( sin ( a ) - 1 )} : sin^{2} (a) + 1

\frac{( sin ^{2} ( a ) + 1 ) ( sin ( a ) + 1 ) ( sin ( a ) - 1 )}{( sin ( a ) + 1 ) ( sin ( a ) - 1 )}

Streiche die gemeinsamen Faktoren:

sin (a) + 1

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

Streiche die gemeinsamen Faktoren:

sin (a) - 1

= sin^{2} (a) + 1

= sin^{2} (a) + 1

= sin^{2} (a) + 1

Verwende die Pythagoreische Identität:

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

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

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

Vereinfache

1 - cos^{2} (a) + 1 : - cos^{2} (a) + 2

1 - cos^{2} (a) + 1

Fasse gleiche Terme zusammen

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

Addiere die Zahlen:

1 + 1 = 2

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

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

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

= 2 - cos^{2} (a)

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

\Rightarrow Wahr

Beliebte Beispiele

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