Not sure what kind of results you are getting but... that
looks correct. Off the cuff, the only other thing that I can think
of that might cause some issues is if you have any other variables
that share the same name.
If so make sure you explicitly define the scope in your
conditional. i.e. isDefined("variables.element") and
variables.element eq "person" etc.
Always scope your variables. Your code will run quicker
(because ColdFusion wont have to check what it is first), be easier
to read and cause you less problems in the long run.
In addition to the suggestions above, I think you want single
quotes around 'element'.
I would tend to do this like so:
<cfif IsDefined('element')>
<CFIF #trim(element)# eq 'person'>
Do something when it equals 'person'
<CFELSE>
What to do if it does not equal 'person'
</CFIF>
<CFELSE>
What to do if 'element' is not defined
</CFIF>