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\frac{a^{2}}{\left(a-1\right)\left(a+1\right)}-\frac{a}{a+1}
Factor a^{2}-1.
\frac{a^{2}}{\left(a-1\right)\left(a+1\right)}-\frac{a\left(a-1\right)}{\left(a-1\right)\left(a+1\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-1\right)\left(a+1\right) and a+1 is \left(a-1\right)\left(a+1\right). Multiply \frac{a}{a+1} times \frac{a-1}{a-1}.
\frac{a^{2}-a\left(a-1\right)}{\left(a-1\right)\left(a+1\right)}
Since \frac{a^{2}}{\left(a-1\right)\left(a+1\right)} and \frac{a\left(a-1\right)}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{a^{2}-a^{2}+a}{\left(a-1\right)\left(a+1\right)}
Do the multiplications in a^{2}-a\left(a-1\right).
\frac{a}{\left(a-1\right)\left(a+1\right)}
Combine like terms in a^{2}-a^{2}+a.
\frac{a}{a^{2}-1}
Expand \left(a-1\right)\left(a+1\right).