(1-1/t^2) / (1/t-t)

(1-1/t^2)/(1/t-t) = -1/t     get common denominators on top and bottom

(  [t^2 - 1] / t^2  )   / (  [ 1 - t^2] / t )   =       factor a -1 out of the numerator on the right

( - [ 1 - t^2] / t^2 )  / ( [ 1 - t^2] / t )  =

( - [ 1 - t^2] / t^2 )  * ( t / [ 1 - t^2] )   =

( - [ 1 - t^2] / [ 1 - t^2] ) * ( t / t^2 )  =

- t  / t^2   =

-1 / t

=  ( t^2 -1)/t^2   / (1-t^2)/t

= (t^2-1)/t^2   x   t/(1-t^2)  = (t^2 -1)/t  x   1/(1-t^2)     Multiply by  -1/-1

(1-t^2)/(-t)  x 1/(1-t^2) =   - 1/t