"Dealing with arrogant people takes a lot of patience and a great deal of self-control. Learning to identify the people who have this trait and avoiding them as much as possible will protect your own reputation, whereas confronting the arrogant person may only make you look worse."
What works for me is I skip the small talk with such people, treat them with respect, remain calm and keep my conversations brief and to the point with them.
Below are some tips and pointers that might help you.
How to Deal With an Arrogant Co-worker Who Drives You Up the Wall:
1) Look Within Yourself
First, let’s talk about you. It’s one thing to be occasionally annoyed
or aggravated by an arrogant co-worker. As long as it’s not frequent
and isn’t disrupting your life, you don’t need to be too concerned.
It’s another thing, though, to become increasingly agitated or
absolutely consumed with frustration and anger. If you have trouble
concentrating, lose sleep, or find yourself complaining frequently and
vehemently about the offending co-worker, you have to figure out how
to overcome your feelings before they wreck your career...
If you feel completely at peace with who you are, then another
person’s quirks should simply be amusing or, at worst, a bit
aggravating—not maddening, and certainly not consuming. So be honest
with yourself about why Braggy Obnoxious bothers you. You don’t have
to tell anyone else about it, but you need to understand it. Your
awareness is your secret weapon, which leads to my next
1) Don’t Hand Over Your Power
Reacting poorly to a person’s arrogance won’t change anything about
it. In fact, if your arrogant colleague—or any problematic colleague,
for that matter—figures out how to push your buttons, you’ve handed
over some of your power. If that person is both arrogant and evil, he
just might use that knowledge to antagonize you and then sit back and
watch you combust. Then you end up looking like the jerk, while your
colleague continues being as arrogant as ever.
Instead of reacting, strive to understand why a person gets to you so
much, which can help you regain control and refocus your energy.
3) Minimize Your Time Together and Keep Things Focused
Even when you have the inner peace of Po the Kung-fu Panda, you’ll
probably still feel some annoyance with certain personality types. You
are human, after all, and you aren’t likely to click with everyone.
Working on a project with that person? Set meetings with strict start
and end times. Do your best to butt those meetings up against other
meetings so you must leave at the pre-determined end time.
Remember, you can only control yourself. You cannot control other
people—no matter how much you may want to—but you can allow them to
control you if you hand your energy and attention over to them. Focus
on yourself, and minimize your time with problematic colleagues. You
will be calmer, and it will show in the quality of your work.
(Source of above points)