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beweisen cot^2(a)-cos^2(a)=cot^2(a)cos^2(a)

Lösung

beweisen

cot^{2} (a) - cos^{2} (a) = cot^{2} (a) cos^{2} (a)

Wahr

Schritte zur Lösung

cot^{2} (a) - cos^{2} (a) = cot^{2} (a) cos^{2} (a)

Manipuliere die linke Seite

cot^{2} (a) - cos^{2} (a)

Drücke mit sin, cos aus

- cos^{2} (a) + cot^{2} (a)

Verwende die grundlegende trigonometrische Identität:

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

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

Vereinfache

- cos^{2} (a) + (\frac{cos ( a )}{sin ( a )})^{2} : \frac{- cos ^{2} ( a ) sin ^{2} ( a ) + cos ^{2} ( a )}{sin ^{2} ( a )}

- cos^{2} (a) + (\frac{cos ( a )}{sin ( a )})^{2}

Wende Exponentenregel an:

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

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

Wandle das Element in einen Bruch um:

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

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

Da die Nenner gleich sind, fasse die Brüche zusammen.:

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

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

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

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

Faktorisiere

\frac{cos ^{2} ( a ) - cos ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )} : \frac{cos ^{2} ( a ) ( 1 - sin ^{2} ( a ) )}{sin ^{2} ( a )}

\frac{cos ^{2} ( a ) - cos ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )}

Faktorisiere

cos^{2} (a) - cos^{2} (a) sin^{2} (a) : cos^{2} (a) (1 - sin^{2} (a))

cos^{2} (a) - cos^{2} (a) sin^{2} (a)

Klammere gleiche Terme aus

cos^{2} (a)

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

= \frac{cos ^{2} ( a ) ( - sin ^{2} ( a ) + 1 )}{sin ^{2} ( a )}

= \frac{( 1 - sin ^{2} ( a ) ) cos ^{2} ( a )}{sin ^{2} ( a )}

Umschreiben mit Hilfe von Trigonometrie-Identitäten

\frac{( 1 - sin ^{2} ( a ) ) cos ^{2} ( a )}{sin ^{2} ( a )}

Verwende die Pythagoreische Identität:

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

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

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

cos^{2} (a) cos^{2} (a) = cos^{4} (a)

cos^{2} (a) cos^{2} (a)

Wende Exponentenregel an:

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

cos^{2} (a) cos^{2} (a) = cos^{2 + 2} (a)

= cos^{2 + 2} (a)

Addiere die Zahlen:

2 + 2 = 4

= cos^{4} (a)

= \frac{cos ^{4} ( a )}{sin ^{2} ( a )}

= \frac{cos ^{4} ( a )}{sin ^{2} ( a )}

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \frac{cos ^{3} ( a )}{sin ( a )} \cdot \frac{cos ( a )}{sin ( a )}

Verwende die grundlegende trigonometrische Identität:

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

\frac{cos ^{3} ( a ) cot ( a )}{sin ( a )}

= cos^{2} (a) cot (a) \frac{cos ( a )}{sin ( a )}

Verwende die grundlegende trigonometrische Identität:

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

cos^{2} (a) cot (a) cot (a)

Vereinfache

cos^{2} (a) cot (a) cot (a) : cos^{2} (a) cot^{2} (a)

cos^{2} (a) cot (a) cot (a)

Wende Exponentenregel an:

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

cot (a) cot (a) = cot^{1 + 1} (a)

= cos^{2} (a) cot^{1 + 1} (a)

Addiere die Zahlen:

1 + 1 = 2

= cos^{2} (a) cot^{2} (a)

cos^{2} (a) cot^{2} (a)

cos^{2} (a) cot^{2} (a)

= cot^{2} (a) cos^{2} (a)

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

\Rightarrow Wahr

p ro v e cos^{2} (θ) (sec^{2} (θ) - 1) = sin^{2} (θ)

p ro v e cot^{2} (y) sec^{2} (y) = 1 + cot^{2} (y)

p ro v e csc (θ) - cot (θ) \cdot cos (θ) = sin (θ)

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

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