Let's solve the expression:

\[
\frac{\left(\frac{a^2 - 1}{a + 1}\right)}{\left(\frac{a}{a^2 - 1}\right)} \cdot \frac{1}{\left(a - \frac{1}{a}\right)}
\]

### Step 1: Simplify the first fraction

The expression inside the first fraction can be simplified. We know that:

\[
a^2 - 1 = (a - 1)(a + 1)
\]

So, the first fraction becomes:

\[
\frac{(a - 1)(a + 1)}{a + 1}
\]

Canceling the common term \((a + 1)\):

\[
= a - 1
\]

Now, we have:

\[
\frac{a - 1}{\left(\frac{a}{(a^2 - 1)}\right)} \cdot \frac{1}{\left(a - \frac{1}{a}\right)}
\]

### Step 2: Simplify the second fraction

We can rewrite the denominator:

\[
\frac{a}{a^2 - 1} = \frac{a}{(a - 1)(a + 1)}
\]

Now the expression becomes:

\[
\frac{a - 1}{\frac{a}{(a - 1)(a + 1)}} \cdot \frac{1}{a - \frac{1}{a}}
\]

### Step 3: Simplify the main fraction

We can simplify the first part by multiplying by the reciprocal of the denominator:

\[
\left(a - 1\right) \cdot \frac{(a - 1)(a + 1)}{a} = \frac{(a - 1)^2(a + 1)}{a}
\]

So now the expression becomes:

\[
\frac{(a - 1)^2(a + 1)}{a} \cdot \frac{1}{a - \frac{1}{a}}
\]

### Step 4: Simplify the second term

We need to simplify \(a - \frac{1}{a}\). This expression can be rewritten as:

\[
a - \frac{1}{a} = \frac{a^2 - 1}{a}
\]

### Step 5: Substitute and simplify

Now substitute back into the main expression:

\[
\frac{(a - 1)^2(a + 1)}{a} \cdot \frac{a}{a^2 - 1}
\]

Since \(a^2 - 1 = (a - 1)(a + 1)\), we can simplify the product:

\[
= \frac{(a - 1)^2(a + 1)}{a} \cdot \frac{a}{(a - 1)(a + 1)}
\]

Cancel out \((a - 1)(a + 1)\) and the \(a\)'s, and we are left with:

\[
a - 1
\]

### Final Answer:

\[
a - 1
\]

Would you like more details or have any questions?

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Here are some related questions to expand on this topic:

1. How can you simplify rational expressions with higher powers of \(a\)?
2. What happens if you change \(a^2 - 1\) to a different polynomial?
3. How would the steps change if we had a more complex denominator in the original problem?
4. How does factoring help in simplifying rational expressions?
5. Can you explain the concept of multiplying by the reciprocal in detail?

**Tip:** Always look for common factors in rational expressions—they simplify the problem significantly!

Learn how to simplify the rational expression ((a^2-1)/(a+1))/(a/(a^2-1)) ∙ 1/(a-1/a) by factoring and using the reciprocal property. This step-by-step guide walks through algebraic simplifications suitable for Grades 9-11, focusing on rational expressions and factoring.

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