Grade 11 1998 Problem 1

Dokažite da za svaki trokut sa stranicama $a$, $b$, $c$ i nasuprotnim kutovima $\alpha$, $\beta$, $\gamma$ vrijedi jednakost $$\left(\frac{b}{c} + \frac{c}{b}\right)\cos\alpha + \left(\frac{c}{a} + \frac{a}{c}\right)\cos\beta + \left(\frac{a}{b} + \frac{b}{a}\right)\cos\gamma = 3.$$