Working with Tangents

Geometry Level 5

\[ \large \dfrac { \sin x }{ \cos y } +\dfrac { \sin y }{ \cos x } =1, \ \ \ \ \ \dfrac { \cos x }{ \sin y } +\dfrac { \cos y }{ \sin x } =6 \]

Let \(x,y\) be real numbers that satisfy the equations above.

Suppose that \(\dfrac { \tan x }{ \tan y } +\dfrac { \tan y }{ \tan x } =\dfrac{p}{q}\) for relatively prime positive integers \(p\) and \(q\). Find \(p+q\).

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