Question 393007
The expression could be written as {{{(2sint)/(cost)^2 + 1/(cost)^2 = (2sint + 1)/(cost)^2}}}.
Find out first the values of t where the top is equal to the bottom, i.e., {{{2sint  + 1 = (cost)^2}}}:
{{{2sint  + 1 = 1 - (sint)^2}}}
==> {{{2sint  = -(sint)^2}}}, or sint(2 + sint) = 0.
==> sint = 0  ==> t =0, +/-{{{pi }}}, +/- {{{2pi }}}, +/- {{{3pi}}},...
At these t values the bottom, {{{(cost)^2}}} is not equal to zero.  The top, {{{2sint + 1}}} is also not equal to 0 (it is equal to 1).  Therefore there are no "holes" in the graph. 
There are vertical asymptotes at t values where {{{(cos t)^2}}} = 0, namely
 t = +/-{{{pi/2}}}, +/- {{{3pi/2 }}}, +/- {{{5pi/2}}},...
The x-intercepts are located at t ={{{7pi/6}}} +/-{{{2n*pi }}} and t ={{{11pi/6}}} +/- {{{2n*pi }}}.  (These are the t values where 2sint  + 1 = 0).