8.3 #43

I definitely had to go back and re-try this problem several times before getting the right solution, so here are some hints in case you get stuck:

cos3x = cos(2x +x) = cos2xcosx - sin2xsinx

sin2x = 2cosxsinx

cos2x= 2(cos^2)x - 1

If you substitute these expressions for sin2x and cos2x, it makes simplifying much easier, because you can factor cosx out of both terms. (I tried the other expressions that also equal cos2x and they only make the problem more complicated.)

Also, later on in the simplifying process, it might help to remember that (cos^2)x-(sin^2)x = cos2x.

Hopefully this might help a little.