Answer
$-\dfrac{2 \sqrt 3}{3}$
Work Step by Step
When $P$ is the period of $f(x)$ then we have
$ f(x+p)=f(x)$
Here, we have $ \csc (-420^{\circ})=-\csc 420^{\circ}=-\csc 60^{\circ} =-\dfrac{1}{\sin 60^{\circ}}$
and $-\dfrac{1}{\sin 60^{\circ}}=-\dfrac{1}{\sqrt 3/2}$
or, $=-\dfrac{2}{\sqrt 3}$
or, $=-\dfrac{2 \sqrt 3}{3}$
