Dividing by variable when solving algebraic equation

[GrandizerII]

1=sqrt(1)=sqrt(-1 * -1)=sqrt(-1)*sqrt(-1)= i*i=-1

The error is in the first equality.
sqrt(1) = +-1 

When you later split the factors you are explicitly forcing one branch of the solution by again, reducing a choice:
sqrt(-1) = +-i

So in full:

+-1 = sqrt(1) = sqrt(-1 * -1) = sqrt(-1) * sqrt(-1) =  +-i * +-i = +-1

Square root of a number, by convention, is always positive.