Solve (1/a+1-2a/a^2-1)(1/a-1)

$$( \frac { 1 } { a + 1 } - \frac { 2 a } { a ^ { 2 } - 1 } ) ( \frac { 1 } { a } - 1 )$$

Evaluate

$$\frac{1}{a}$$

Short Solution Steps

Factor a^{2}-1.

$$\left(\frac{1}{a+1}-\frac{2a}{\left(a-1\right)\left(a+1\right)}\right)\left(\frac{1}{a}-1\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+1 and \left(a-1\right)\left(a+1\right) is \left(a-1\right)\left(a+1\right). Multiply \frac{1}{a+1} times \frac{a-1}{a-1}.

$$\left(\frac{a-1}{\left(a-1\right)\left(a+1\right)}-\frac{2a}{\left(a-1\right)\left(a+1\right)}\right)\left(\frac{1}{a}-1\right)$$

Since \frac{a-1}{\left(a-1\right)\left(a+1\right)} and \frac{2a}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a-1-2a}{\left(a-1\right)\left(a+1\right)}\left(\frac{1}{a}-1\right)$$

Combine like terms in a-1-2a.

$$\frac{-a-1}{\left(a-1\right)\left(a+1\right)}\left(\frac{1}{a}-1\right)$$

Extract the negative sign in -a-1.

$$\frac{-\left(a+1\right)}{\left(a-1\right)\left(a+1\right)}\left(\frac{1}{a}-1\right)$$

Cancel out a+1 in both numerator and denominator.

$$\frac{-1}{a-1}\left(\frac{1}{a}-1\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a}{a}.

$$\frac{-1}{a-1}\left(\frac{1}{a}-\frac{a}{a}\right)$$

Since \frac{1}{a} and \frac{a}{a} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-1}{a-1}\times \frac{1-a}{a}$$

Multiply \frac{-1}{a-1} times \frac{1-a}{a} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-\left(1-a\right)}{\left(a-1\right)a}$$

Extract the negative sign in 1-a.

$$\frac{-\left(-1\right)\left(a-1\right)}{a\left(a-1\right)}$$

Cancel out a-1 in both numerator and denominator.

$$\frac{-\left(-1\right)}{a}$$

Multiply -1 and -1 to get 1.

$$\frac{1}{a}$$

Expand

$$\frac{1}{a}$$

Short Solution Steps

Factor a^{2}-1.

$$\left(\frac{1}{a+1}-\frac{2a}{\left(a-1\right)\left(a+1\right)}\right)\left(\frac{1}{a}-1\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+1 and \left(a-1\right)\left(a+1\right) is \left(a-1\right)\left(a+1\right). Multiply \frac{1}{a+1} times \frac{a-1}{a-1}.

$$\left(\frac{a-1}{\left(a-1\right)\left(a+1\right)}-\frac{2a}{\left(a-1\right)\left(a+1\right)}\right)\left(\frac{1}{a}-1\right)$$

Since \frac{a-1}{\left(a-1\right)\left(a+1\right)} and \frac{2a}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a-1-2a}{\left(a-1\right)\left(a+1\right)}\left(\frac{1}{a}-1\right)$$

Combine like terms in a-1-2a.

$$\frac{-a-1}{\left(a-1\right)\left(a+1\right)}\left(\frac{1}{a}-1\right)$$

Extract the negative sign in -a-1.

$$\frac{-\left(a+1\right)}{\left(a-1\right)\left(a+1\right)}\left(\frac{1}{a}-1\right)$$

Cancel out a+1 in both numerator and denominator.

$$\frac{-1}{a-1}\left(\frac{1}{a}-1\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a}{a}.

$$\frac{-1}{a-1}\left(\frac{1}{a}-\frac{a}{a}\right)$$

Since \frac{1}{a} and \frac{a}{a} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-1}{a-1}\times \frac{1-a}{a}$$

Multiply \frac{-1}{a-1} times \frac{1-a}{a} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-\left(1-a\right)}{\left(a-1\right)a}$$

Extract the negative sign in 1-a.

$$\frac{-\left(-1\right)\left(a-1\right)}{a\left(a-1\right)}$$

Cancel out a-1 in both numerator and denominator.

$$\frac{-\left(-1\right)}{a}$$

Multiply -1 and -1 to get 1.

$$\frac{1}{a}$$