Answer
$1$
Work Step by Step
$\csc{(\theta+ 2 \pi k}) = \csc{\theta}$
$\therefore \csc{\dfrac{9 \pi}{2}} = \csc{\left(\dfrac{\pi}{4}+ (2 \pi \times 2)\right)} = \csc{\dfrac{\pi}{2}}$
$ \csc{\dfrac{\pi}{2}} = \boxed{1}$
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