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