Another identity, brain still not working

Ok, I know it seems like I am spunging, but there are thirty of these on this assignement, only stumped on two -- so far. The identity is:

(sin(x) + sin(3x))/(cos(x) + cos(3x) = tan(2x)

The only identities I have covered so far in my book are double-angle, half-angle, sum and difference, and the basics (reciprocal, quotentient, pythagorean).

I have tried subbing the tan(2x) for 2tan(x)/(1 - tan^2(x)), but I can't figure out where to go from there. I also tried to work on the other side -- sin(2x + x), use the sum identity, etc. -- but got really confused. I am doing trig on a distance learning course, and I don't really have access to a teacher or fellow students to talk it over with, so I really appreciate the help! I have been working on these way too long.