Ratio of Trigonometric Functions tanx/tany = 1/3;  sin2x/sin2y = 3/4 where 0 < x,y< pi/2. What is the Value of 
tan2x/tan2y
I couldn't find the appropriate trig identities to use for this problem, and what relationship for sin and tan that we can exploit.

$$\frac{\tan x}{\tan y}=\frac{1}{3};\;\frac{\sin 2x}{\sin 2y}=\frac{3}{4}$$
set $\tan x =t;\;\tan y=u$
$$\sin 2x=\frac{2t}{1+t^2};\;\sin 2y=\frac{2u}{1+u^2}$$
$$\frac{t}{u}=\frac{1}{3}\to u=3t$$
$$\frac{2t}{1+t^2}\frac{1+u^2}{2u}=\frac{3}{4}$$
$$\frac{2t}{1+t^2}\frac{1+9t^2}{6t}=\frac{3}{4}$$
$$8t(1+9t^2)=18t(1+t^2)\to 8t(1+9t^2)-18t(1+t^2)=0\to 2t(4+36t^2-9-9t^2)=0$$
$$\;t=\pm\sqrt{\frac{5}{27}};\;u=\pm 3\sqrt{\frac{5}{27}}=\pm\sqrt{\frac{5}{3}}$$
it $t=0$ then $u=0$ which is impossible since $u$ is the denominator of the fraction
if $t=\pm\sqrt{\frac{5}{27}};\;x=\pm\arctan\sqrt{\frac{5}{27}}+k\pi,\;\forall k\in\mathbb{Z}$
$u=\pm\sqrt{\frac{5}{3}};\;y=\pm\arctan \sqrt{\frac{5}{3}}+h\pi ,\;\forall h\in\mathbb{Z}$
