What is a word for purposely trying to avoid answering a question? [duplicate]

I thought you were looking for a verb. If you're still looking for a verb:

Prevaricate, evade, dodge.

http://dictionary.reference.com/browse/evade?s=t

Consider, fudge.

: (v.i.) to avoid coming to grips with something : to fudge on an issue.

: (v.t.) to avoid coming to grips with (a subject, issue, etc.); evade; dodge Random House

You can also circumvent a question.

circumvent: to avoid or try to avoid fulfilling, answering, or performing The Free Dictionary

What you describe sounds like dodging the question:

Question dodging is the intentional avoidance of answering a question.

(https://en.wikipedia.org/wiki/Question_dodging)

In the context of politics, another word for this kind of trick would be pivot (it's a noun, but I think you can verb it):

Brett O'Donnell is a debate consultant who trains Republican candidates. He has worked with George W. Bush and John McCain, and for a short time earlier this year, he helped prep Mitt Romney.

O'Donnell is an expert on "the pivot."

If you have watched a debate, you have watched a pivot. "The pivot is a way of taking a question that might be on a specific subject, and moving to answer it on your own terms," O'Donnell says.

(http://www.npr.org/2012/10/03/162103368/how-politicians-get-away-with-dodging-the-question)

Pivoting to answer the question "on your own terms" may result in all kinds of shenanigans: obfuscation, digressing, changing the subject, etc.