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beweisen csc^4(x)-cot^4(x)=2cot^2(x)+1

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

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

Faktorisiere

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

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

Schreibe

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

um:

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

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

Wende Exponentenregel an:

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

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

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

Wende Exponentenregel an:

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

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

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

Faktorisiere

csc^{2} (x) - cot^{2} (x) : (csc (x) + cot (x)) (csc (x) - cot (x))

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

= (csc (x) + cot (x)) (csc (x) - cot (x))

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Multipliziere aus

(csc (x) + cot (x)) (csc (x) - cot (x)) : csc^{2} (x) - cot^{2} (x)

(csc (x) + cot (x)) (csc (x) - cot (x))

Wende Formel zur Differenz von zwei Quadraten an:

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

a = csc (x), b = cot (x)

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

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

Verwende die Pythagoreische Identität:

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

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

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

Vereinfache

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

Verwende die Pythagoreische Identität:

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

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

Vereinfache

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

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

Fasse gleiche Terme zusammen

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

Addiere gleiche Elemente:

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

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

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

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

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

\Rightarrow Wahr

p ro v e cos (- x) - sin (- x) = cos (x) + sin (x)

p ro v e cos (θ) sec (θ) = 1

p ro v e cos (a + b) + cos (a - b) = 2 cos (a) cos (b)

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

p ro v e tan (\frac{π}{2} - u) = cot (u)