$\alpha  = \frac{\pi }{2}{\sin ^2}\theta  \Rightarrow \tan \alpha  + \cot (\frac{\pi }{2} - \alpha ) = 2\sqrt 3  \Rightarrow 2\tan \alpha  = 2\sqrt 3 $
$ \Rightarrow \tan \alpha  = \sqrt 3 \xrightarrow{{0 \lt \alpha  \lt \frac{\pi }{2}}}\alpha  = \frac{\pi }{3} \Rightarrow \frac{\pi }{2}{\sin ^2}\theta  = \frac{\pi }{3} \Rightarrow {\sin ^2}\theta  = \frac{2}{3}$
$ \Rightarrow {\tan ^2}\theta  = 2,{\cot ^2}\theta  = \frac{1}{2} \Rightarrow \sin (5\frac{\pi }{3} \times 2) + \cos (\frac{{5\pi }}{3} \times \frac{1}{2}) = $
$\sin (3\pi  + \frac{\pi }{3}) + \cos (\pi  - \frac{\pi }{6}) =  - \sin \frac{\pi }{3} - \cos \frac{\pi }{6} =  - \frac{{\sqrt 3 }}{2} - \frac{{\sqrt 3 }}{2} =  - \sqrt 3 $