2013 AIME II Problem 15

Let A,B,C be angles of an acute triangle with \begin{aligned} \cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\ \cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9} \end{aligned} There are positive integers p, q, r, and s for which \cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \frac{p-q\sqrt{r}}{s}, where p+q and s are relatively prime and r is not divisible by the square of any prime. Find p+q+r+s.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.