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beweisen cos^2(x)+cos^2(x)cot^2(x)=cot^2(x)

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

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

Drücke mit sin, cos aus

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

Verwende die grundlegende trigonometrische Identität:

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

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

Vereinfache

cos^{2} (x) + cos^{2} (x) (\frac{cos ( x )}{sin ( x )})^{2} : \frac{cos ^{2} ( x ) sin ^{2} ( x ) + cos ^{4} ( x )}{sin ^{2} ( x )}

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

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

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

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

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

Wende Exponentenregel an:

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

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

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

Multipliziere Brüche:

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

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

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

cos^{2} (x) cos^{2} (x)

Wende Exponentenregel an:

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

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

= cos^{2 + 2} (x)

Addiere die Zahlen:

2 + 2 = 4

= cos^{4} (x)

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

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

Wandle das Element in einen Bruch um:

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

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

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

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

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

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

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

Faktorisiere

\frac{cos ^{4} ( x ) + cos ^{2} ( x ) sin ^{2} ( x )}{sin ^{2} ( x )} : \frac{cos ^{2} ( x ) ( cos ^{2} ( x ) + sin ^{2} ( x ) )}{sin ^{2} ( x )}

\frac{cos ^{4} ( x ) + cos ^{2} ( x ) sin ^{2} ( x )}{sin ^{2} ( x )}

Faktorisiere

cos^{4} (x) + cos^{2} (x) sin^{2} (x) : cos^{2} (x) (cos^{2} (x) + sin^{2} (x))

cos^{4} (x) + cos^{2} (x) sin^{2} (x)

Wende Exponentenregel an:

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

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

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

Klammere gleiche Terme aus

cos^{2} (x)

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

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die Pythagoreische Identität:

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

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

Vereinfache

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die grundlegende trigonometrische Identität:

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

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

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

Verwende die grundlegende trigonometrische Identität:

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

cot (x) cot (x)

Vereinfache

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

cot (x) cot (x)

Wende Exponentenregel an:

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

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

= cot^{1 + 1} (x)

Addiere die Zahlen:

1 + 1 = 2

= cot^{2} (x)

cot^{2} (x)

cot^{2} (x)

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

\Rightarrow Wahr

p ro v e 2 cos (x - \frac{π}{4}) = cos (x) + sin (x)

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

p ro v e cos (x - \frac{3 π}{2}) = - sin (x)

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

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