Answer
$-\dfrac {\sqrt {2}}{2} $
Work Step by Step
Period of $cos$ function is $2\pi$
It means
$\cos \alpha =\cos \left( 2\pi +\alpha \right) =\cos \left( -2\pi +\alpha \right) $
$\cos \left( 585\right) =\cos \left( 360+225\right) =\cos \left( 2\pi +\dfrac {5\pi }{4}\right) =\cos (\dfrac {5\pi }{4})=\cos \left( \pi +\dfrac {\pi }{4}\right) =\cos \pi \cos \dfrac {\pi }{4}-\sin \pi \sin \dfrac {\pi }{4}=\left( -1\right) \times \cos \dfrac {\pi }{4}-0=-\cos \dfrac {\pi }{4}=-\dfrac {\sqrt {2}}{2} $
