erweitern (1+sqrt(1-x^2))(sqrt(1+(x^2)/(1-x^2)))

Lösung

erweitern

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

\frac{1}{1 - x ^{2}} + 1

Schritte zur Lösung

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

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

Wende das Distributivgesetz an:

a (b + c) = ab + a c

a = (1 + \frac{x ^{2}}{1 - x ^{2}}), b = 1, c = 1 - x^{2}

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

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

Vereinfache

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

= \frac{1}{1 - x ^{2}} + 1

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