Simplify sec^2(t) / sec^2(t)-1 to an expression involving a si for trig function with no fractions.

Consider this:
sec t=1/ cos t
cos² t=(1-sin²t)
1/sin t=csc t

therefore:
sec²t / (sec² t-1)=(1/cos² t) / [(1/cos²t)  -1]=(1/cos² t) / [(1-cos²t)/cos²t]=
=[1/(1-sin²t)] / [(1-(1-sin²t)) / (1-sin²t)]=[1 / (1-sin²t)] / [(sin²t) / (1-sin²t)]=
=1/sin²t=csc ² t

sec²t / (sec²t-1)=csc²t