<s>[INST] Trigonometry Problems Let $x$ and $y$ be real numbers such that $\frac{\sin x}{\sin y} = 3$ and $\frac{\cos x}{\cos y} = \frac{1}{2}$. The value of $\frac{\sin 2x}{\sin 2y} + \frac{\cos 2x}{\cos 2y}$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
My student left me with a list of questions just now (he also said that he wanted to be helped at school :P)
 [/INST] The first two equations, squared, allow you to solve for $\sin^2$ and $\sin^2y$ as follows:
$$\begin{cases}\sin^2x&=9\sin^2y,\\4(1-\sin^2x)&=1-\sin^2y.\end{cases}$$
Then $\sin^2x=\dfrac{27}{35},\sin^2y=\dfrac3{35}$.
From this, by the double angle formula
$$\begin{cases}\cos 2x=-\dfrac{19}{35},\\\cos 2y=\dfrac{29}{35}.\end{cases}$$ 
The rational ratio is $$\dfrac pq=\frac32-\frac{19}{29}=\frac{49}{58}.$$
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