10. If $\frac{\sin x}{\sin y}=3$ and $\frac{\cos x}{\cos y}=\frac{1}{2}$, find $\frac{\sin 2 x}{\sin 2 y}+\frac{\cos 2 x}{\cos 2 y}$.
(1 mark)若 $\frac{\sin x}{\sin y}=3$ 及 $\frac{\cos x}{\cos y}=\frac{1}{2}$, 求 $\frac{\sin 2 x}{\sin 2 y}+\frac{\cos 2 x}{\cos 2 y}$ 。
( 1 分)
10. $\frac{49}{58}$
10. From the given equations we have $4 \sin ^{2} x=36 \sin ^{2} y$ and $4 \cos ^{2} x=\cos ^{2} y$. It follows that
$$
4=4 \sin ^{2} x+4 \cos ^{2} x=36 \sin ^{2} y+\cos ^{2} y=36 \sin ^{2} y+\left(1-\sin ^{2} y\right)
$$
and so $\sin ^{2} y=\frac{3}{35}$. In the same way we get $\sin ^{2} x=\frac{27}{35}, \cos ^{2} x=\frac{8}{35}$ and $\cos ^{2} y=\frac{32}{35}$. Hence
$$
\frac{\sin 2 x}{\sin 2 y}+\frac{\cos 2 x}{\cos 2 y}=\frac{2 \sin x \cos x}{2 \sin y \cos y}+\frac{\cos ^{2} x-\sin ^{2} x}{\cos ^{2} y-\sin ^{2} y}=2(3)\left(\frac{1}{2}\right)+\frac{\frac{8}{35}}{\frac{32}{35}-\frac{27}{35}}=\frac{49}{58} .
$$