Working with Tangents

Geometry Level 5

sinxcosy+sinycosx=1,     cosxsiny+cosysinx=6 \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,yx,y be real numbers that satisfy the equations above.

Suppose that tanxtany+tanytanx=pq\dfrac { \tan x }{ \tan y } +\dfrac { \tan y }{ \tan x } =\dfrac{p}{q} for relatively prime positive integers pp and qq. Find p+qp+q.

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