What's wrong with this 1 = -1 proof?

Question

Stared at this proof for 10 minutes, perhaps even more. Still a quite stumped, but I'm pretty sure the answer is staring me right in the face.

Okay, so we know that $i^2 = -1$.

Dividing both sides by $i$:

$$i = - \frac{1}{i}$$

Squaring both sides:

$$i^2 = -\frac{1^2}{i^2}$$

Obviously $i^2 = -1$, as previously shown, so therefore:

$$-1 = -\frac{1}{-1}$$

Both negatives become a positive, so we're left with:

$$-1 = \frac{1}{1}$$

Which simplifies to:

$$-1 = 1$$

I'm not quite sure what's wrong here. Unless I'm there's an important step I skipped, I don't really see any problem here.

Answer 1

The square of $-x$ is $x^2$, not $-(x^2)$. Your error is in the "squaring both sides" step.

Answer 2

$\left(\frac{-1}{i}\right)^2=\frac{(-1)^2}{i^2}=\frac{1}{-1}=-1$

Answer 3

You're doing mistake in the "squaring both side" step. You're squaring like this -(1/i)²

It should be done like this (-1/i)²