Question

If `2x = sqrt a +1/sqrt a` then what is the value of `sqrt(x^2 - 1)/{x - sqrt(x^2 - 1)}` ?

Answer 1

I tried by writing an expression for x only:
`x=1/2×(sqrt(a)+1/(sqrt(a)))`
I then substituted it in the expression:
`(sqrt(x^2-1))/(x-sqrt(x^2-1))`
Considering that:
`sqrt(x^2-1)=sqrt(1/4×(sqrt(a)+1/(sqrt(a)))^2-1)=`
`=sqrt((a^2-2a+1)/(4a)=`
`=(a-1)/(2sqrt(a))`

So that `(sqrt(x^2-1))/(x-sqrt(x^2-1))` becomes:

`(a-1)/(2sqrt(a))/(sqrt(a)/2+1/(2sqrt(a))-(a-1)/(2sqrt(a)))=`
`=(a-1)/2`