Originally Posted by a-j

This one is fairly easy, when you take square roots you need to assume both values that the input could have produced (i.e. sqrt(1) = -1,1). So your reasoning should have been something like:

sqrt(-1/1)=sqrt(1/-1)
sqrt(-1)/sqrt(1)=sqrt(1)/sqrt(-1)
(±i)/(±1)=(±1)/(±i) => i^2=±1 (if you work out all combinations)
-1=1, or -1=-1 (the latter is correct)

When people end up with something like 2=1, it doesn't mean they actually proved 2=1, it means there is a contradiction which means the they did something incorrect or their initial conditions were wrong. Proof by contradiction is very strong tool if used correctly. However I understand this is all in fun.