can somebody prove (1 - E^(-2/x))/(1 - E^(-1/x)) equals to E^(-1/x)+1 where E is exponential?

{1 - E^(-2/x)} / [1 - E^(-1/x)]

= {1^2 - [E^(-1/x)]^2} / [1 - E^(-1/x)]

= [1 + E^(-1/x)] [1 - E^(-1/x)] / [1 - E^(-1/x)]

= 1 + E^(-1/x)

{[1 - (e^(-2/x))]/[1 - (e^(-1/x))]} * {[1 + (e^(-1/x))]/[1 + (e^(-1/x))]}

[1 - (e^(-2/x))] * [1 + (e^(-1/x))]/[1 - (e^(-2/x))]; from rule (e^a)(e^b) = e^(a + b)

[1 + (e^(-1/x))]

Proof:
[1-e^(-2/x)]/[1-e^(-1/x)]=

[1-1/e^(2/x)]/[1-1/e^(1/x)]=

[e^(2/x)-1][e^(1/x)]/
[e^(2/x)][e^(1/x)-1]=

[e^(1/x)+1][e^(1/x)-1]/
[e^(1/x)][e^(1/x)-1]=

[e^(1/x)+1]/e^(1/x)=

1+e^(-1/x)