Regarding usage of $\pm$ sign I want to say that $x=1$ or $x=-1$, but we don't know which one (or we could possibly calculate it, but it is complicated). Anyhow, for my future proof all I need to know is that $x$ is either $1$ or $-1$. 
Is it OK to write it as $x=\pm 1$ ?
I am asking because it seems that meaning here is different from for example when we use $\pm$ in the formula of roots of quadratic  equations.  
If I can't use $\pm$, then is there some other way I can denote it?
 A: In identities like $\tan\left(\dfrac\pi4\pm\dfrac\theta2\right)=\sec\theta\pm\tan\theta$, the notation means that if the first is "$+$" then so is the second and if the first is "$-$" then so is the second.
In things like $\cos(\alpha\pm\beta) = \cos\alpha\cos\beta \mp \sin\alpha \sin\beta$, if the first is "$+$" then the second is "$-$" and vice-versa.
If you say that a poll shows that $40\%\pm2\%$ of voters will vote "yes" in next month's referendum, it might mean that those are the bounds of a $90\%$, or maybe $95\%$ confidence interval.  I've been known to write things like
$$
\text{The bounds of a $90\%$ confidence interval are } 122\pm6.97.
$$
If one were to write something like $x\in\{\pm1\}$, it might in some contexts reasonably be construed as meaning $x\in\{1,-1\}$.
A: In general writing that $x = \pm 1$ means that $x = 1$ or $x = -1$. So if that is what you want to write, technically you can do that.
For example if you are asked to solve the equation $x^{2} - 1= 0 $, then you could write that $x^2 - 1 = 0$ implies that $x=\pm 1$.
Indeed, if you don't like writing things like $x =\pm 1$ then you can just write $x = 1$ or $x = -1$. And if you are writing a proof out carefully and you want to make sure that there is no confusion, then I would personally say to write it out in words. So it if perfectly fine to write things like "we know therefore that $x=1$ or that $x = -1$". Or you can write things like" We want to show that $x =1 $ or $x = -1$". If you want to avoid using word, you can also write $x = 1 \lor x = -1$ also meaning that $x$ is equal to $1$ or $x$ is equal to $-1$.
A: You might just like to say $|x|=1$. Often when a sign isn't important, it's perfectly acceptable to think in terms of magnitude.
Of course, if the sign carries forward in the proof but in the end that it doesn't matter, you might like to say "without loss of generality, let $x=+1$ with the negative case being proved analogously".
A: I often write $x\in\{\pm 1\}$. But I think that $x=\pm 1$ is also OK and unambiguous.
