Question

How to simplify `(sqrt [1/x^2-1]-1/x) * ((1-x) / [sqrt (1-x^2)-1+x] + sqrt(1+x) / (sqrt(1+x)-sqrt(1-x)))` ?

Answer 1

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

Explanation:

Observe that `sqrt(1-x^2)-1+x=sqrt(1-x^2)-(1-x)`

= `sqrt(1-x)(sqrt(1+x)-sqrt(1-x))`, and hence

`(sqrt(1/x^2-1)-1/x)((1-x)/(sqrt(1-x^2)-1+x)+sqrt(1+x)/ (sqrt(1+x)-sqrt(1-x)))`

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

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

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

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

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

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

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

= `1/x^2(1-x^2-1)`

= `-x^2/x^2`

= `-1`