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العربية

beweisen 1-(1-cos^2(x))/(1-cos(x))=-cos(x)

Lösung

beweisen

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

Wahr

Schritte zur Lösung

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

Manipuliere die linke Seite

1 - \frac{1 - cos ^{2} ( x )}{1 - cos ( x )}

Vereinfache

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

1 - \frac{1 - cos ^{2} ( x )}{1 - cos ( x )}

Streiche

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

\frac{1 - cos ^{2} ( x )}{1 - cos ( x )}

Faktorisiere

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

1 - cos^{2} (x)

Klammere gleiche Terme aus

- 1

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

Faktorisiere

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

cos^{2} (x) - 1

Schreibe

1

um:

1^{2}

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

= (cos (x) + 1) (cos (x) - 1)

= - (cos (x) + 1) (cos (x) - 1)

= - \frac{( cos ( x ) + 1 ) ( cos ( x ) - 1 )}{1 - cos ( x )}

Streiche

- \frac{( cos ( x ) + 1 ) ( cos ( x ) - 1 )}{1 - cos ( x )} : cos (x) + 1

- \frac{( cos ( x ) + 1 ) ( cos ( x ) - 1 )}{1 - cos ( x )}

- cos (x) + 1 = - (cos (x) - 1)

= - \frac{( cos ( x ) + 1 ) ( cos ( x ) - 1 )}{- ( cos ( x ) - 1 )}

Fasse zusammen

= \frac{( cos ( x ) + 1 ) ( cos ( x ) - 1 )}{cos ( x ) - 1}

Streiche die gemeinsamen Faktoren:

cos (x) - 1

= cos (x) + 1

= cos (x) + 1

= 1 - (cos (x) + 1)

- (cos (x) + 1) : - cos (x) - 1

- (cos (x) + 1)

Setze Klammern

= - (cos (x)) - (1)

Wende Minus-Plus Regeln an

+ (- a) = - a

= - cos (x) - 1

= 1 - cos (x) - 1

Vereinfache

1 - cos (x) - 1 : - cos (x)

1 - cos (x) - 1

Fasse gleiche Terme zusammen

= - cos (x) + 1 - 1

1 - 1 = 0

= - cos (x)

= - cos (x)

= - cos (x)

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

\Rightarrow Wahr

p ro v e \frac{sin ( 4 θ ) + sin ( 2 θ )}{cos ( 4 θ ) + cos ( 2 θ )} = tan (3 θ)

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

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

p ro v e (cot (X) - csc (X)) (cot (X) + csc (X)) = - 1

p ro v e \frac{2 sin ( 2 θ ) cos ( θ )}{2 sin ( 2 θ )} = cos (θ)