Question: I've started with thinking that if X and Y are perfectly correlated, then it is the same as looking at the correlation of Y and -Y (since the X provides no new information) and thus the correlation is -1.

Is this correct?

Answer: 
I'll treat this as [self-study](/questions/tagged/self-study "show questions tagged 'self-study'"), and I'd encourage you to read [its wiki](https://stats.stackexchange.com/tags/self-study/info) and add the tag.

Your argument is already very good. Here are a few pointers. Feel free to write a comment so we can discuss and work towards a good answer.

* I assume you are looking at Pearson's correlation, right? (Does your argument work for other measures of correlation?)
* What does a perfect correlation mean *graphically*?
* What will $X+Y$ and $X-Y$ look like *graphically* if $X$ and $Y$ are perfectly Pearson-correlated?
If $X$ is a linear function of $Y$ (definition of perfect correlation), then both $X-Y$ and $X+Y$ will be linear functions of $Y$, and therefore are linear functions of each other. 

So, $X-Y$ and $X+Y$ are perfectly correlated.