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))) ?

1 Answer
Apr 24, 2018

(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