IIT JEE Trigonometry 2014 - Given equations involving sides find ratio of in/circumcircle radius

In this trigonometry problem from the 2014 IIT JEE paper, we are given some equations involving the sides of the triangle and we have to find the ratio of the incircle radius with the circumcircle radius. The problem requires recalling some formulas for the incircle and circumcircle radius. We also show how to derive these formulas.