a)

$\frac{\sin {120}^{\circ }+\cos {135}^{\circ }}{2\mathrm{tg}{150}^{\circ }}=\frac{\sin \left({90}^{\circ }+{30}^{\circ }\right)+\cos \left({90}^{\circ }+{45}^{\circ }\right)}{2\mathrm{tg}\left({180}^{\circ }-{30}^{\circ }\right)}=\frac{\cos {30}^{\circ }-\sin {45}^{\circ }}{-2\mathrm{tg}{30}^{\circ }}=\frac{\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}}{-2\cdot \frac{\sqrt{3}}{3}}=-\frac{\frac{\sqrt{3}-\sqrt{2}}{2}}{\frac{2\sqrt{3}}{3}}=-\frac{\sqrt{3}-\sqrt{2}}{2}\cdot \frac{3}{2\sqrt{3}}=-\frac{3\sqrt{3}-3\sqrt{2}}{4\sqrt{3}}=-\frac{9-3\sqrt{6}}{12}=-\frac{3-\sqrt{6}}{4}=\frac{\sqrt{6}-3}{4}$  

 

b)

$\frac{{\sin }^2{135}^{\circ }+{\sin }^2{45}^2}{\cos {45}^{\circ }-\cos {135}^{\circ }}=\frac{{\sin }^2\left({90}^{\circ }+{45}^{\circ }\right)+{\sin }^2{45}^{\circ }}{\cos {45}^{\circ }-\cos \left({90}^{\circ }+{45}^{\circ }\right)}=\frac{\stackrel{=1}{\overbrace{{\cos }^2{45}^{\circ }+{\sin }^2{45}^{\circ }}}}{\cos {45}^{\circ }+\sin {45}^{\circ }}=\frac{1}{\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$