In a triangle PQR, angle PQR = θ and angle QPR = 2θ. Prove that cos θ = p/(2q).

This is meant to be solved without using the trig identity sin 2θ = 2 sin θ cos θ, but it is intended for the sine rule to be used. Using the sine rule, I have got as far as

p / sin 2θ = q / sin θ

but I can't get any further. Help much appreciated!