I would like to use the following commands (in order to simplify writing):
\newcommand{\<}{\langle} \renewcommand{\>}{\rangle} Would there be any dangerous consequences of this? I know that by default \> inserts \medmuskip, but I am not using this command.
You risk that some obscure package defines something in term of \> or define \< and \> themselves.
A quick search reveals that \< is defined in
amsdtx.cls amsldoc.cls circ.sty cjhebrew.sty gmmeta.sty greektonoi.sty jlreq.cls lhcyralt.sty lhcyrkoi.sty lhcyrwin.sty pax.sty AlProTex.sty dev209.sty The command \> is redefined in
bxcjkjatype.sty gn-logic14.sty greektonoi.sty lhcyralt.sty lhcyrkoi.sty lhcyrwin.sty pax.sty AlProTex.sty sgamevar.sty zxjatype.sty I can't exclude there are other instances where different ways of defining commands are used.
I'd not do it anyway, even if it's not that risky. Better do
\usepackage{mathtools} \DeclarePairedDelimiters{\foo}{\langle}{\rangle} (choose a meaningful name for your application) and use
\foo{x} \foo[\big]{x} \foo[\Big]{x} \foo[\bigg]{x} \foo[\Bigg]{x} \foo*{x} for choosing the size of the delimiters (the last one is for automatic sizing with \left and \right, use it carefully).
If you decide to use \< and \> beware that inside a tabbing environment you will have to use \langle and \rangle because tabbing redefines the two commands for its own purposes (but just inside the environment).
\DeclarePairedDelimiters, thanks for pointing out this interesting command.
\>and\<might cause havoc if you need to use thetabbingenvironment.$\<a\>$in a tabbing environment would cause havoc. If this is needed, it's sufficient to use\langleand\ranglein the special situation. Nothing would happen otherwise.