Answer
$-\dfrac {1}{2}$
Work Step by Step
$csc^{-1}x=\cos ^{-1}\dfrac {1}{x};$
$\cos \left( \cos ^{-1}x\right) =x$
$\Rightarrow \cos \left( csc-1\left( -2\right) \right) =\cos \left( \cos ^{-1}\left( \dfrac {1}{-2}\right) \right) =-\dfrac {1}{2}$
