Question

How do you simplify `(1/sqrt(a-1)+sqrt(a+1))/(1/sqrt(a+1)-1/sqrt(a-1))div sqrt(a+1)/((a-1)sqrt(a+1)-(a+1)sqrt(a-1)),` a>1 ?

Answer 1

Huge math formatting...

Explanation:

`color(blue)(((1/sqrt(a-1)+sqrt(a+1))/(1/sqrt(a+1)-1/sqrt(a-1)))/ (sqrt(a+1)/((a-1)sqrt(a+1)-(a+1)sqrt(a-1)))`

`=color(red)(((1/sqrt(a-1)+sqrt(a+1))/((sqrt(a-1)-sqrt(a+1))/(sqrt(a+1) cdot sqrt(a-1))))/ (sqrt(a+1)/(sqrt(a-1) cdot sqrt(a-1) cdot sqrt(a+1)-sqrt(a+1) cdot sqrt(a+1)sqrt(a-1)))`

`=color(blue)(((1/sqrt(a-1)+sqrt(a+1))/((sqrt(a-1)-sqrt(a+1))/(sqrt(a+1) cdot sqrt(a-1))))/ (sqrt(a+1)/(sqrt(a+1) cdot sqrt(a-1)(sqrt(a-1)-sqrt(a+1)))`

`=color(red)((1/sqrt(a-1)+sqrt(a+1))/((sqrt(a-1)-sqrt(a+1))/(sqrt(a+1) cdot sqrt(a-1)))xx(sqrt(a+1) cdot sqrt(a-1)(sqrt(a-1)-sqrt(a+1)))/sqrt(a+1)`

`=color(blue)((1/sqrt(a-1)+sqrt(a+1))xx((sqrt(a+1) cdot sqrt(a-1))/(sqrt(a-1)-sqrt(a+1)))xx(cancel((sqrt(a+1))) cdot sqrt(a-1)(sqrt(a-1)-sqrt(a+1)))/cancelsqrt(a+1))`

`=color(red)(((1+sqrt(a+1) cdot sqrt(a-1))/(sqrt(a-1)))xx((sqrt(a+1) cdot sqrt(a-1))/(sqrt(a-1)-sqrt(a+1)))xx sqrt(a-1)cdot(sqrt(a-1)-sqrt(a+1))`

`=color(blue)(((1+sqrt(a+1) cdot sqrt(a-1))/cancel(sqrt(a-1)))xx((sqrt(a+1) cdot cancel((sqrt(a-1))))/color(red)(cancel(color(green)((sqrt(a-1)-sqrt(a+1)))))xx sqrt(a-1)cdot color(red)(cancel color(green)((sqrt(a-1)-sqrt(a+1)))`

`=color(red)(ul(bar(|color(blue)((1+sqrt(a+1) cdot sqrt(a-1))cdot(sqrt((a+1)(a-1))))|`

Answer 2

`sqrt(a^2-1)+a^2-1`

Explanation:

To simplify things greatly we will use `u^2=a+1` and `v^2=a-1`, which gives us:
`(v^-1+u)/(u^-1-v^-1)*(uv^2-vu^2)/u=((v^-1+u)(uv^2-vu^2))/(u(u^-1-v^-1))=(uv-u^2+(uv)^2-vu^3)/(1-uv^-1)=(uv(1+uv)-u^2(1+uv))/((v-u)/v)=(uv(1+uv)(v-u))/(v-u)=uv(1+uv)`

`uv(1+uv)=uv+u^2v^2=sqrt(a-1)sqrt(a+1)+(a-1)(a+1)=sqrt(a^2-1)+a^2-1`