Does Anyone Know how to Simplify the Following Expression?

[jinydu]

x - sqrt(x^2 - 1)

Thanks

[antiroach]

as x gets very large you get 0 :)

[jinydu]

I mean, an exact simplification.

[cheesehead]

This is almost certainly not what you want, and is not exactly a simplification, but "Factoris" at http://wims.unice.fr/~wims/wims.cgi?...ebra/factor.en gives:

The factorization of x-(x²-1)^0.5 =
(0 - i) + x + (0 + 0.5i) x² + (0.0 + 0.125i) x⁴ + (0.0 + 0.0625i) x⁶ + O(x⁸)

;-)

[rogue]

It appears to be something out of trigonometry, so I looked it up and found:

arccosh(x) = ln(x - sqrt(x² - 1))

so

e^arccosh(x) = x - sqrt(x² - 1)

[jinydu]

Thanks for the help, but rogue's formula doesn't work for x = 2.0996011

[rogue]

I don't have a calculator that can do arccosh. What do both terms evaluate to?

[jinydu]

LHS ~= 3.945765984

RHS ~= 0.2534362159

[rogue]

Oddly, the website I found this in has '-', but others have '+'. Using '+' instead of '-', you get

1) arccosh(x) = ln(x + sqrt(x^2 - 1))

2) e^arccosh(x) = x + sqrt(x^2 - 1)

3) x - sqrt(x^2 - 1) = (x - sqrt(x^2 - 1)) * (x + sqrt(x^2 - 1))/(x + sqrt(x^2 - 1))

4) x - sqrt(x^2 - 1) = (x^2 - (x^2 - 1))/(x + sqrt(x^2 - 1))

5) x - sqrt(x^2 - 1) = 1/(x + sqrt(x^2 - 1))

6) x - sqrt(x^2 - 1) = 1/(e^arccosh(x))

7) x - sqrt(x^2 - 1) = e^-arccosh(x)

I suspect that e^-arccosh(x) might be simplified as well, but I'll leave that for the next person.

[cheesehead]

Originally Posted by jinydu

Thanks for the help, but rogue's formula doesn't work for x = 2.0996011

. . .

LHS ~= 3.945765984

RHS ~= 0.2534362159

Did you forget to try the negative square root on the RHS? :)
Using that one, my calculator shows LHS = RHS = 3.94576598405592039632709017264425

Alternatively, you can try the negative arccosh with the positive square root, in which case LHS = RHS = 0.253436215944079603672909827355751