Question 1209533
You're asking to prove that if *a* and *b* are real numbers, then:

| (a - b) / (√(1 - a²) - √(1 - b²)) | ≤ √((1 - a²)(1 - b²)) / |ab|

This inequality is *not generally true* for all real numbers *a* and *b*.  It has restrictions and can be quite complex to analyze directly.  It's also unclear what the intended restrictions might be, as the given inequality isn't valid for all a and b.

Here's why and what we *can* say:

1. **Domain Restrictions:**  The square roots require 1 - a² ≥ 0 and 1 - b² ≥ 0. This means -1 ≤ a ≤ 1 and -1 ≤ b ≤ 1.

2. **Denominator Issues:** The denominator √(1 - a²) - √(1 - b²) can be zero if a = b, which would make the left side undefined.  We'd need to exclude the case where a = b.  Also, if a and b are close, the denominator can be very small, making the left side very large.

3. **Counterexample:** Let a = 0.5 and b = 0.6.
   * Left side: |(0.5 - 0.6) / (√(1 - 0.5²) - √(1 - 0.6²))| = |(-0.1) / (√0.75 - √0.64)| ≈ 0.41
   * Right side: √(0.75 * 0.64) / |0.5 * 0.6| ≈ 1.63
   In this case, the inequality holds. However, this doesn't prove it for all a and b.

4. **What We *Can* Say (and a related inequality):**

A related and *true* inequality (under the domain restrictions mentioned above) is derived from the Mean Value Theorem and involves trigonometric substitution. If we let a = cos(x) and b = cos(y) where x and y are in the interval \[0, π], then the expression inside the absolute value can be related to sin(c)(cos(x) - cos(y)) for some c between x and y.

It's important to note that the inequality you provided is not a standard or commonly known inequality.  If you encountered it in a specific context (a textbook, a problem set, etc.), there might be additional conditions or context that would make it valid.  Without that context, it's likely incorrect as stated.

**In summary:** The inequality you provided is not generally true for all real numbers *a* and *b*. It has domain restrictions, a potential for division by zero, and can be shown to be false with counterexamples.  A related inequality involving trigonometric substitutions does exist but is different from the one you provided.