Помогите! Рассчитайте выражение (даю 30 балов)

Решите задачу:

$\frac{1}{\sqrt3}\cdot ctg\frac{4\pi}{3}+2\cdot cos\frac{7\pi}{3}= \frac{1}{\sqrt3}ctg(\pi +\frac{\pi}{3})+2\cdot cos(2\pi +\frac{\pi }{3} )=\\\\= \frac{1}{\sqrt3}\cdot ctg\frac{\pi }{3}+2\cdot cos\frac{\pi }{3}= \frac{cos\frac{\pi}{3}}{\sqrt3\cdot sin\frac{\pi}{3}}+2cos\frac{\pi }{3}=\\\\=\frac{cos\frac{\pi}{3}+2\cdot \sqrt3\cdot sin\frac{\pi}{3}\cdot cos\frac{\pi}{3}}{\sqrt3\cdot sin\frac{\pi}{3}}= \frac{cos\frac{\pi}{3}\cdot (1+2\sqrt3\cdot sin\frac{\pi}{3})}{\sqrt3\cdot sin\frac{\pi}{3}} =$

$= \frac{1}{\sqrt3} \cdot ctg\frac{\pi }{3}\cdot (1+2\sqrt3\cdot sin\frac{\pi}{3})= \frac{1}{\sqrt3} \cdot \frac{\sqrt3}{3}\cdot (1+2\sqrt3\cdot \frac{\sqrt3}{2})=\\\\=\frac{1}{3}\cdot (1+3)=\frac{4}{3}$