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beweisen tan^4(t)+tan^2(t)=sec^4(t)-sec^2(t)

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

tan^{4} (t) + tan^{2} (t)

Faktorisiere

tan^{2} (t) + tan^{4} (t) : tan^{2} (t) (1 + tan^{2} (t))

tan^{2} (t) + tan^{4} (t)

Wende Exponentenregel an:

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

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

= tan^{2} (t) + tan^{2} (t) tan^{2} (t)

Klammere gleiche Terme aus

tan^{2} (t)

= tan^{2} (t) (1 + tan^{2} (t))

= (1 + tan^{2} (t)) tan^{2} (t)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

(1 + tan^{2} (t)) tan^{2} (t)

Verwende die Pythagoreische Identität:

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

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

= (1 + sec^{2} (t) - 1) (sec^{2} (t) - 1)

Vereinfache

1 + sec^{2} (t) - 1 : sec^{2} (t)

1 + sec^{2} (t) - 1

Fasse gleiche Terme zusammen

= sec^{2} (t) + 1 - 1

1 - 1 = 0

= sec^{2} (t)

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

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

Multipliziere aus

(- 1 + sec^{2} (t)) sec^{2} (t) : - sec^{2} (t) + sec^{4} (t)

(- 1 + sec^{2} (t)) sec^{2} (t)

= sec^{2} (t) (- 1 + sec^{2} (t))

Wende das Distributivgesetz an:

a (b + c) = ab + a c

a = sec^{2} (t), b = - 1, c = sec^{2} (t)

= sec^{2} (t) (- 1) + sec^{2} (t) sec^{2} (t)

Wende Minus-Plus Regeln an

+ (- a) = - a

= - 1 \cdot sec^{2} (t) + sec^{2} (t) sec^{2} (t)

Vereinfache

- 1 \cdot sec^{2} (t) + sec^{2} (t) sec^{2} (t) : - sec^{2} (t) + sec^{4} (t)

- 1 \cdot sec^{2} (t) + sec^{2} (t) sec^{2} (t)

1 \cdot sec^{2} (t) = sec^{2} (t)

1 \cdot sec^{2} (t)

Multipliziere:

1 \cdot sec^{2} (t) = sec^{2} (t)

= sec^{2} (t)

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

sec^{2} (t) sec^{2} (t)

Wende Exponentenregel an:

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

sec^{2} (t) sec^{2} (t) = sec^{2 + 2} (t)

= sec^{2 + 2} (t)

Addiere die Zahlen:

2 + 2 = 4

= sec^{4} (t)

= - sec^{2} (t) + sec^{4} (t)

= - sec^{2} (t) + sec^{4} (t)

= - sec^{2} (t) + sec^{4} (t)

= sec^{4} (t) - sec^{2} (t)

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

\Rightarrow Wahr

p ro v e tan (A) = \frac{sec ( A )}{csc ( A )}

p ro v e csc (- x) = - csc (x)

p ro v e sin (β) + cos (β) cot (β) = csc (β)

p ro v e sin (θ) cos (θ) = tan (θ)

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