$a)\sin \left(-{855}^o\right)=-\sin {885}^o=-\sin \left({720}^o+135\right)=-\sin {135}^o=-\sin \left({180}^o-{45}^o\right)=-\sin {45}^o=-\frac{\sqrt{2}}{2}$  

$b)\cos {2490}^o=\cos \left({2160}^o+{330}^o\right)=\cos \left(6\cdot {360}^o+{330}^o\right)=\cos \left({330}^o\right)=\cos \left({360}^o-{30}^o\right)=\cos {30}^o=\frac{\sqrt{3}}{2}$  

$c)\mathrm{tg}{570}^o=\mathrm{tg}\left({540}^o+{30}^o\right)=\mathrm{tg}{30}^o=\frac{\sqrt{3}}{3}$  

 

$d)\mathrm{ctg}\left(-{450}^o\right)=-\mathrm{ctg}\left({450}^o\right)=-\mathrm{ctg}\left({360}^o+{90}^o\right)=-\mathrm{ctg}{90}^o=-0=0$  

 

$e)\sin \left(\frac{25}{4}\pi \right)=\sin \left(\frac{24}{4}\pi +\frac{1}{4}\pi \right)=\sin \left(6\pi +\frac{1}{4}\pi \right)=\sin \left(3\cdot 2\pi +\frac{1}{4}\pi \right)=\sin \left(\frac{1}{4}\pi \right)=\frac{\sqrt{2}}{2}$  

 

$f)\cos \left(-\frac{37}{6}\pi \right)=\cos \left(\frac{37}{6}\pi \right)=\cos \left(\frac{36}{6}\pi +\frac{1}{6}\pi \right)=\cos \left(3\cdot 2\pi +\frac{1}{6}\pi \right)=\cos \left(\frac{1}{6}\pi \right)=\frac{\sqrt{3}}{2}$  

 

$g)\mathrm{tg}\left(-\frac{51}{4}\pi \right)=-\mathrm{tg}\left(\frac{51}{4}\pi \right)=-\mathrm{tg}\left(\frac{48}{4}\pi +\frac{3}{4}\pi \right)=-\mathrm{tg}\left(12\cdot \pi +\frac{3}{4}\pi \right)=-\mathrm{tg}\left(\frac{3}{4}\pi \right)=$  

$=-\mathrm{tg}\left(\pi -\frac{1}{4}\pi \right)=-\left(-\mathrm{tg}\left(\frac{1}{4}\pi \right)\right)=\mathrm{tg}\left(\frac{1}{4}\pi \right)=1$  

 

$h)\mathrm{ctg}\frac{13}{6}\pi =\mathrm{ctg}\left(\frac{12}{6}\pi +\frac{1}{6}\pi \right)=\mathrm{ctg}\left(2\cdot \pi +\frac{1}{6}\pi \right)=\mathrm{ctg}\frac{1}{6}\pi =\sqrt{3}$