Each step below seems reasonable, but obviously there is a problem.

\[\frac{1}{-1}\] = \[\frac{-1}{1}\]

If two numbers are equal, their square roots are equal:

\[\sqrt{\frac{1}{-1}}\] = \[\sqrt{\frac{-1}{1}}\]

The square root of a quotient is the quotient of the square roots:

\[\frac{\sqrt{1}}{\sqrt{-1}}\] = \[\frac{\sqrt{-1}}{\sqrt{1}}\]

Simplifying:

\[\frac{1}{j}\] = \[\frac{j}{1}\]

Cross-mulitiplying:

1\[^{2}\] = j\[^{2}\]

Therefore: 1 = -1