For the complex variable 'z', show that the following equation is true.

Question

anonymous · Answer

$$\frac{i}{2}(z^2-z^{-2})=-\sin(2\theta)$$

anonymous · Answer

It's in a textbook chapter about Euler's formula. I would only need a brief explanation and then I will probably get it.

anonymous · Answer

A brief explanation of how to do it that is, not Euler's formula

zepdrix · Answer

The question is a little weird.
I think we have to assume that $\Large\rm \|z\|=1$ in order for this to work out.

$$\Large\rm z^2=\left(e^{\mathcal i \theta}\right)^2=e^{2\mathcal i \theta}=\cos(2\theta)+\mathcal i \sin(2\theta)$$

If magnitude z isn't 1, we get r's floating around... which is weird >.<

anonymous · Answer

Hmmm, I see, I will try and do it with that, thanks. I just looked back at the question and it doesn't say anything about the magnitude of z :/

zepdrix · Answer

Well normally z written in polar form would be something like: $\Large\rm z=r e^{\mathcal i \theta}$
So we would end up with r's :d
Maybe I'm just being picky though lol

anonymous · Answer

Well, looks like you're right, by substituting in values the equation doesn't hold true when the magnitude of 'z' is larger than 1. What a stupid question. It becomes almost trivial when you make that assumption.
Thanks :)