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Beliebt Trigonometrie >

beweisen (1-tan^4(x))/(sec^2(x))=1-tan^2(x)

Lösung

beweisen

\frac{1 - tan ^{4} ( x )}{sec ^{2} ( x )} = 1 - tan^{2} (x)

Lösung

Wahr

Schritte zur Lösung

\frac{1 - tan ^{4} ( x )}{sec ^{2} ( x )} = 1 - tan^{2} (x)

Manipuliere die linke Seite

\frac{1 - tan ^{4} ( x )}{sec ^{2} ( x )}

Umschreiben mit Hilfe von Trigonometrie-Identitäten

\frac{1 - tan ^{4} ( x )}{sec ^{2} ( x )}

Verwende die Pythagoreische Identität:

sec^{2} (x) = tan^{2} (x) + 1

= \frac{1 - tan ^{4} ( x )}{tan ^{2} ( x ) + 1}

Vereinfache

\frac{1 - tan ^{4} ( x )}{tan ^{2} ( x ) + 1} : - (tan (x) + 1) (tan (x) - 1)

\frac{1 - tan ^{4} ( x )}{tan ^{2} ( x ) + 1}

Faktorisiere

1 - tan^{4} (x) : - (tan^{2} (x) + 1) (tan (x) + 1) (tan (x) - 1)

1 - tan^{4} (x)

Klammere gleiche Terme aus

- 1

= - (tan^{4} (x) - 1)

Faktorisiere

tan^{4} (x) - 1 : (tan^{2} (x) + 1) (tan (x) + 1) (tan (x) - 1)

tan^{4} (x) - 1

Schreibe

tan^{4} (x) - 1

um:

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

tan^{4} (x) - 1

Schreibe

1

um:

1^{2}

= tan^{4} (x) - 1^{2}

Wende Exponentenregel an:

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

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

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

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

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

Faktorisiere

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

tan^{2} (x) - 1

Schreibe

1

um:

1^{2}

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

= (tan (x) + 1) (tan (x) - 1)

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

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

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

Streiche die gemeinsamen Faktoren:

tan^{2} (x) + 1

= - (tan (x) + 1) (tan (x) - 1)

= - (tan (x) + 1) (tan (x) - 1)

= - (tan (x) + 1) (tan (x) - 1)

Multipliziere aus

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

- (- 1 + tan (x)) (1 + tan (x))

Multipliziere aus

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

(- 1 + tan (x)) (1 + tan (x))

Wende Formel zur Differenz von zwei Quadraten an:

(a - b) (a + b) = a^{2} - b^{2}

a = tan (x), b = 1

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

Wende Regel an

1^{a} = 1

1^{2} = 1

= tan^{2} (x) - 1

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

Setze Klammern

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

Wende Minus-Plus Regeln an

- (- a) = a, - (a) = - a

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

= 1 - tan^{2} (x)

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

\Rightarrow Wahr

Beliebte Beispiele

beweisen 3-sec^2(x)=2-tan^2(x)

p ro v e 3 - sec^{2} (x) = 2 - tan^{2} (x)

beweisen (cot(x)+tan(x))^2=csc^2(x)sec^2(x)

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

beweisen cos(3x)=4cos(3x)-3cos(x)

p ro v e cos (3 x) = 4 cos (3 x) - 3 cos (x)

beweisen sec^2(a)+csc^2(a)=sec^2(a)csc^2(a)

p ro v e sec^{2} (a) + csc^{2} (a) = sec^{2} (a) csc^{2} (a)

beweisen (2sin(x)+cos(x))^2+(2cos(x)-sin(x))^2=5

p ro v e (2 sin (x) + cos (x))^{2} + (2 cos (x) - sin (x))^{2} = 5