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beweisen cot^4(x)+cot^2(x)=csc^4(x)-csc^2(x)

Lösung

beweisen

cot^{4} (x) + cot^{2} (x) = csc^{4} (x) - csc^{2} (x)

Wahr

Schritte zur Lösung

cot^{4} (x) + cot^{2} (x) = csc^{4} (x) - csc^{2} (x)

Manipuliere die rechte Seite

csc^{4} (x) - csc^{2} (x)

Faktorisiere

- csc^{2} (x) + csc^{4} (x) : csc^{2} (x) (- 1 + csc^{2} (x))

- csc^{2} (x) + csc^{4} (x)

Wende Exponentenregel an:

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

csc^{4} (x) = csc^{2} (x) csc^{2} (x)

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

Klammere gleiche Terme aus

csc^{2} (x)

= csc^{2} (x) (- 1 + csc^{2} (x))

= (- 1 + csc^{2} (x)) csc^{2} (x)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

(- 1 + csc^{2} (x)) csc^{2} (x)

Verwende die Pythagoreische Identität:

csc^{2} (x) = 1 + cot^{2} (x)

= (- 1 + 1 + cot^{2} (x)) (1 + cot^{2} (x))

Vereinfache

= cot^{2} (x) (1 + cot^{2} (x))

= cot^{2} (x) (1 + cot^{2} (x))

Multipliziere aus

(1 + cot^{2} (x)) cot^{2} (x) : cot^{2} (x) + cot^{4} (x)

(1 + cot^{2} (x)) cot^{2} (x)

= cot^{2} (x) (1 + cot^{2} (x))

Wende das Distributivgesetz an:

a (b + c) = ab + a c

a = cot^{2} (x), b = 1, c = cot^{2} (x)

= cot^{2} (x) \cdot 1 + cot^{2} (x) cot^{2} (x)

= 1 \cdot cot^{2} (x) + cot^{2} (x) cot^{2} (x)

Vereinfache

1 \cdot cot^{2} (x) + cot^{2} (x) cot^{2} (x) : cot^{2} (x) + cot^{4} (x)

1 \cdot cot^{2} (x) + cot^{2} (x) cot^{2} (x)

1 \cdot cot^{2} (x) = cot^{2} (x)

1 \cdot cot^{2} (x)

Multipliziere:

1 \cdot cot^{2} (x) = cot^{2} (x)

= cot^{2} (x)

cot^{2} (x) cot^{2} (x) = cot^{4} (x)

cot^{2} (x) cot^{2} (x)

Wende Exponentenregel an:

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

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

= cot^{2 + 2} (x)

Addiere die Zahlen:

2 + 2 = 4

= cot^{4} (x)

= cot^{2} (x) + cot^{4} (x)

= cot^{2} (x) + cot^{4} (x)

= cot^{2} (x) + cot^{4} (x)

= cot^{4} (x) + cot^{2} (x)

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

\Rightarrow Wahr

p ro v e 1 - tanh^{2} (x) = sech^{2} (x)

p ro v e cos (π - x) = - cos (x)

p ro v e \frac{1 + sin ( θ )}{1 - sin ( θ )} - \frac{1 - sin ( θ )}{1 + sin ( θ )} = 4 tan (θ) sec (θ)

p ro v e cos (7 x) + cos (3 x) = 2 cos (5 x) cos (2 x)

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