Answer
$\sqrt 2 $
Work Step by Step
$csc\dfrac {9\pi }{4}=\dfrac {1}{\sin \left( \dfrac {9\pi }{4}\right) }=\dfrac {1}{\sin \left( 2\pi +\dfrac {\pi }{4}\right) }$
Since $sin$ function has a period of $2\pi$
$\sin \left( 2\pi +\alpha \right) =\sin \alpha $
So we get
$csc\dfrac {9\pi }{4}=\dfrac {1}{\sin \left( 2\pi +\dfrac {\pi }{4}\right) }=\dfrac {1}{\sin \dfrac {\pi }{4}}=\dfrac {1}{\dfrac {\sqrt {2}}{2}}=\dfrac {2}{\sqrt {2}}=\sqrt {2} $
