Talk:2013 AIME I Problems/Problem 14

For solution 1 where they talk about $P \sin \theta + Q \cos \theta = \cos \theta - \frac{1}{2} P$ , I got $\cos \theta - \frac{1}{4} \cos \theta + \frac{1}{8} \sin 2 \theta - \frac{1}{16} \cos 3 \theta + \frac{1}{32} \sin 4 \theta - \dots$ instead (the same as in solution 2), which is not what the solution 1 got. If the writer of solution 1 could resolve this, that would be great.