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beweisen tan^2(t)sin^2(t)=tan^2(t)-sin^2(t)

Lösung

beweisen

tan^{2} (t) sin^{2} (t) = tan^{2} (t) - sin^{2} (t)

Wahr

Schritte zur Lösung

tan^{2} (t) sin^{2} (t) = tan^{2} (t) - sin^{2} (t)

Manipuliere die linke Seite

tan^{2} (t) sin^{2} (t)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

tan^{2} (t) sin^{2} (t)

Verwende die Pythagoreische Identität:

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

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

= tan^{2} (t) (1 - cos^{2} (t))

= tan^{2} (t) (1 - cos^{2} (t))

Multipliziere aus

(1 - cos^{2} (t)) tan^{2} (t) : tan^{2} (t) - tan^{2} (t) cos^{2} (t)

(1 - cos^{2} (t)) tan^{2} (t)

= tan^{2} (t) (1 - cos^{2} (t))

Wende das Distributivgesetz an:

a (b - c) = ab - a c

a = tan^{2} (t), b = 1, c = cos^{2} (t)

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

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

Multipliziere:

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

= tan^{2} (t) - tan^{2} (t) cos^{2} (t)

= tan^{2} (t) - cos^{2} (t) tan^{2} (t)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

tan^{2} (t) - cos^{2} (t) tan^{2} (t)

Verwende die grundlegende trigonometrische Identität:

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

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

Vereinfache

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

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

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

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

Wende Exponentenregel an:

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

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

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

Multipliziere Brüche:

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

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

Streiche die gemeinsamen Faktoren:

cos^{2} (t)

= sin^{2} (t)

= tan^{2} (t) - sin^{2} (t)

= tan^{2} (t) - sin^{2} (t)

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

\Rightarrow Wahr

p ro v e 8 sin^{2} (θ) + 9 cos^{2} (θ) = 8 + cos^{2} (θ)

p ro v e sec^{2} (x) + tan^{2} (x) = 1 + 2 tan^{2} (x)

p ro v e cot (x + π) = cot (x)

p ro v e tan (x) (2 - cos (x))^{2} = 0

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