Question 1209533: if a , b tow number \[Element] R , prove that :, |(a - b)/(sqrt(1 - a ^2) - sqrt(1 - b ^2))| < sqrt3
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.
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