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