Question

Question #e9266

Answer 1

After using u= `cot^(-1)(sqrt(1+x)/sqrt(1-x))` transform,

`cotu=sqrt(1+x)/sqrt(1-x)`

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

`(1-x)*(cotu)^2=1+x`

`(cotu)^2-x*(cotu)^2=1+x`

`x*[(cotu)^2+1]=(cotu)^2-1`

`x*(cscu)^2=(cscu)^2*[(cosu)^2-(sinu)^2]`

`x=(cosu)^2-(sinu)^2`

`x=cos2u`

Hence,

`y=(sin[cot^(-1)(sqrt(1+x)/sqrt(1-x))])^2`

=`(sinu)^2`

=`1/2*2(sinu)^2`

=`1/2*(1-cos2u)`

=`(1-x)/2`

Explanation:

1) I used `u=cot^(-1)(sqrt(1+x)/sqrt(1-x))` transform for finding x in terms of u. Finally I found `x=cos2u`.

2) I rewrote `(sinu)^2` in terms of `cos2u`. Finally I found `(sin[cot^(-1)(sqrt(1+x)/sqrt(1-x))])^2` as `(1-x)/2`.