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beweisen (tan^2(x)+1)(cos^2(x)-1)=-tan^2(x)

Lösung

beweisen

(tan^{2} (x) + 1) (cos^{2} (x) - 1) = - tan^{2} (x)

Wahr

Schritte zur Lösung

(tan^{2} (x) + 1) (cos^{2} (x) - 1) = - tan^{2} (x)

Manipuliere die linke Seite

(tan^{2} (x) + 1) (cos^{2} (x) - 1)

Multipliziere aus

(- 1 + cos^{2} (x)) (1 + tan^{2} (x)) : - 1 - tan^{2} (x) + cos^{2} (x) + cos^{2} (x) tan^{2} (x)

(- 1 + cos^{2} (x)) (1 + tan^{2} (x))

Wende Ausklammerungsregel an (VANI):

(a + b) (c + d) = a c + a d + b c + b d

a = - 1, b = cos^{2} (x), c = 1, d = tan^{2} (x)

= (- 1) \cdot 1 + (- 1) tan^{2} (x) + cos^{2} (x) \cdot 1 + cos^{2} (x) tan^{2} (x)

Wende Minus-Plus Regeln an

+ (- a) = - a

= - 1 \cdot 1 - 1 \cdot tan^{2} (x) + 1 \cdot cos^{2} (x) + cos^{2} (x) tan^{2} (x)

Vereinfache

- 1 \cdot 1 - 1 \cdot tan^{2} (x) + 1 \cdot cos^{2} (x) + cos^{2} (x) tan^{2} (x) : - 1 - tan^{2} (x) + cos^{2} (x) + cos^{2} (x) tan^{2} (x)

- 1 \cdot 1 - 1 \cdot tan^{2} (x) + 1 \cdot cos^{2} (x) + cos^{2} (x) tan^{2} (x)

Multipliziere die Zahlen:

1 \cdot 1 = 1

= - 1 - 1 \cdot tan^{2} (x) + 1 \cdot cos^{2} (x) + cos^{2} (x) tan^{2} (x)

Multipliziere:

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

= - 1 - tan^{2} (x) + 1 \cdot cos^{2} (x) + cos^{2} (x) tan^{2} (x)

Multipliziere:

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

= - 1 - tan^{2} (x) + cos^{2} (x) + cos^{2} (x) tan^{2} (x)

= - 1 - tan^{2} (x) + cos^{2} (x) + cos^{2} (x) tan^{2} (x)

= - 1 + cos^{2} (x) - tan^{2} (x) + cos^{2} (x) tan^{2} (x)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

- 1 + cos^{2} (x) - tan^{2} (x) + cos^{2} (x) tan^{2} (x)

Verwende die Pythagoreische Identität:

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

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

= - sin^{2} (x) - tan^{2} (x) + cos^{2} (x) tan^{2} (x)

Verwende die grundlegende trigonometrische Identität:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - sin^{2} (x) - tan^{2} (x) + cos^{2} (x) (\frac{sin ( x )}{cos ( x )})^{2}

Vereinfache

cos^{2} (x) (\frac{sin ( x )}{cos ( x )})^{2} : sin^{2} (x)

cos^{2} (x) (\frac{sin ( x )}{cos ( x )})^{2}

(\frac{sin ( x )}{cos ( x )})^{2} = \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

(\frac{sin ( x )}{cos ( x )})^{2}

Wende Exponentenregel an:

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

= \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

= \frac{sin ^{2} ( x )}{cos ^{2} ( x )} cos^{2} (x)

Multipliziere Brüche:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

Streiche die gemeinsamen Faktoren:

cos^{2} (x)

= sin^{2} (x)

= - sin^{2} (x) - tan^{2} (x) + sin^{2} (x)

Addiere gleiche Elemente:

- sin^{2} (x) + sin^{2} (x) = 0

= - tan^{2} (x)

= - tan^{2} (x)

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

\Rightarrow Wahr

p ro v e csc (θ) cos (θ) = cot (θ)

p ro v e sec (x) = \frac{1}{cos ( x )}

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

p ro v e cot (x) - tan (x) = 2 cot (2 x)

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