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beweisen 1-tan^4(A)=2sec^2(A)-sec^4(A)

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

Manipuliere die rechte Seite

2 sec^{2} (A) - sec^{4} (A)

Faktorisiere

- sec^{4} (A) + 2 sec^{2} (A) : sec^{2} (A) (- sec^{2} (A) + 2)

- sec^{4} (A) + 2 sec^{2} (A)

Wende Exponentenregel an:

a^{b + c} = a^{b} a^{c}

sec^{4} (A) = sec^{2} (A) sec^{2} (A)

= - sec^{2} (A) sec^{2} (A) + 2 sec^{2} (A)

Klammere gleiche Terme aus

sec^{2} (A)

= sec^{2} (A) (- sec^{2} (A) + 2)

= (2 - sec^{2} (A)) sec^{2} (A)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

(2 - sec^{2} (A)) sec^{2} (A)

Verwende die Pythagoreische Identität:

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

= (2 - (tan^{2} (A) + 1)) (tan^{2} (A) + 1)

Multipliziere aus

2 - (tan^{2} (A) + 1) : - tan^{2} (A) + 1

2 - (tan^{2} (A) + 1)

- (tan^{2} (A) + 1) : - tan^{2} (A) - 1

- (tan^{2} (A) + 1)

Setze Klammern

= - (tan^{2} (A)) - (1)

Wende Minus-Plus Regeln an

+ (- a) = - a

= - tan^{2} (A) - 1

= 2 - tan^{2} (A) - 1

Vereinfache

2 - tan^{2} (A) - 1 : - tan^{2} (A) + 1

2 - tan^{2} (A) - 1

Fasse gleiche Terme zusammen

= - tan^{2} (A) + 2 - 1

Addiere/Subtrahiere die Zahlen:

2 - 1 = 1

= - tan^{2} (A) + 1

= - tan^{2} (A) + 1

= (- tan^{2} (A) + 1) (tan^{2} (A) + 1)

= (- tan^{2} (A) + 1) (tan^{2} (A) + 1)

Multipliziere aus

(1 + tan^{2} (A)) (1 - tan^{2} (A)) : 1 - tan^{4} (A)

(1 + tan^{2} (A)) (1 - tan^{2} (A))

Wende Formel zur Differenz von zwei Quadraten an:

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

a = 1, b = tan^{2} (A)

= 1^{2} - (tan^{2} (A))^{2}

Vereinfache

1^{2} - (tan^{2} (A))^{2} : 1 - tan^{4} (A)

1^{2} - (tan^{2} (A))^{2}

Wende Regel an

1^{a} = 1

1^{2} = 1

= 1 - (tan^{2} (A))^{2}

(tan^{2} (A))^{2} = tan^{4} (A)

(tan^{2} (A))^{2}

Wende Exponentenregel an:

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

= tan^{2 \cdot 2} (A)

Multipliziere die Zahlen:

2 \cdot 2 = 4

= tan^{4} (A)

= 1 - tan^{4} (A)

= 1 - tan^{4} (A)

= 1 - tan^{4} (A)

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

\Rightarrow Wahr

p ro v e cos^{4} (θ) - sin^{4} (θ) = 2 cos^{2} (θ) - 1

p ro v e sin (x) tan (x) + cos (x) = \frac{1}{cos ( x )}

p ro v e \frac{1 - cos ( x )}{sin ( x )} = tan (\frac{x}{2})

p ro v e cos (4 θ) = 8 cos^{4} (θ) - 8 cos^{2} (θ) + 1

p ro v e 1 - 2 cos^{2} (x) = 2 sin^{2} (x) - 1