Answer
$1$
Work Step by Step
$$\eqalign{
&\text{Let }I= \int_{\pi /6}^{\pi /2} {\csc t\cot t} dt \cr
& {\text{Integrate using the basic formula }}\int {\csc x\cot x} dx = - \csc x + C \cr
&I = \left[ { - \csc t} \right]_{\pi /6}^{\pi /2} \cr
& {\text{Using the fundamental theorem of calculus}}{\text{, part 2}} \cr
& I=\left[ { - \csc t} \right]_{\pi /6}^{\pi /2} = - \left[ {\csc \left( {\frac{\pi }{2}} \right) - \csc \left( {\frac{\pi }{6}} \right)} \right] \cr
& {\text{Simplify}} \cr
& I=\left[ { - \csc t} \right]_{\pi /6}^{\pi /2} = - \left[ {1 - 2} \right] = 1 \cr} $$