Find the values of the following trigonometric ratios:
$\sin \left(\frac{-11\mathrm{\pi }}{6}\right)$
 

Find the values of the following trigonometric ratio: $cosec\left(\frac{-20\mathrm{\pi }}{3}\right)$
 

Prove that: $\tan 225°\cot 405°+\tan 765°\cot 675°=0$

Prove that: $\sin \frac{8\pi }{3}\cos \frac{23\pi }{6}+\cos \frac{13\pi }{3}\sin \frac{35\pi }{6}=\frac{1}{2}$