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Simplify the expression:

[tex]\[
\frac{x-y}{x^2-1} \cdot \frac{x-1}{x^2-y^2}
\][/tex]

Note: The original text had unnecessary symbols and was partially incoherent. The problem is now focused on simplifying the given mathematical expression.


Sagot :

Sure, let's simplify the expression step-by-step.

We are given the expression:

[tex]\[ \frac{x-y}{x^2-1} \cdot \frac{x-1}{x^2-y^2} \][/tex]

To simplify this expression, we will first factorize the denominators:

1. Factorize [tex]\(x^2 - 1\)[/tex] and [tex]\(x^2 - y^2\)[/tex]:
- [tex]\(x^2 - 1\)[/tex] is a difference of squares and can be factored as [tex]\((x-1)(x+1)\)[/tex].
- [tex]\(x^2 - y^2\)[/tex] is also a difference of squares and can be factored as [tex]\((x-y)(x+y)\)[/tex].

Given these factorizations, we can rewrite the expression as:

[tex]\[ \frac{x-y}{(x-1)(x+1)} \cdot \frac{x-1}{(x-y)(x+y)} \][/tex]

2. Cancel out common terms:
- We notice that [tex]\((x-y)\)[/tex] appears in the numerator of the first fraction and the denominator of the second fraction, so they can be cancelled out.
- Similarly, [tex]\((x-1)\)[/tex] appears in the numerator of the second fraction and the denominator of the first fraction, so they can be cancelled out as well.

Rewriting with the cancelled terms, we have:

[tex]\[ \frac{1}{(x+1)} \cdot \frac{1}{(x+y)} \][/tex]

3. Combine the fractions:

Since we are multiplying two fractions, we simply multiply the numerators together and the denominators together:

[tex]\[ \frac{1 \cdot 1}{(x+1)(x+y)} = \frac{1}{(x+1)(x+y)} \][/tex]

Finally, the simplified form of the given expression is:

[tex]\[ \frac{1}{x^2 + x y + x + y} \][/tex]

So, the answer is:

[tex]\[ \boxed{\frac{1}{x^2 + x y + x + y}} \][/tex]