[tex]\dfrac{\left(\sin \frac{\pi}{4}+\cos \frac{3\pi}{2}\right)\cdot \text{tg}\,\frac{\pi}{3}}{\text{ctg}\,\frac{\pi}{6}-\text{ctg}\,\frac{\pi}{2}} =\dfrac{\left(\frac{\sqrt{2}}{2}+0\right)\cdot \sqrt{3}}{\sqrt{3}-0} =\dfrac{\frac{\sqrt{2}}{2}\cdot \sqrt{3}}{\sqrt{3}}=\boxed{\frac{\sqrt{2}}{2}}[/tex]
