How to eliminate $x,y$ from that system?

Weierstrass substitution makes Eliminate working: Solutions of sys are 2Pi-periodic

sys={a*Tan[x] == b*Tan[y], a*Sin[x]^2 + b*Cos[x]^2 == m, b*Sin[y]^2 +a*Cos[y]^2 == n}

Weierstrass-substitution constraints the solution {x,y} to -Pi<x,y<Pi

sysu = TrigExpand[sys /. {x -> 2 ArcTan[ux], y -> 2 ArcTan[uy]}] // Simplify

cond=Eliminate[sysu, {ux, uy}] // FullSimplify
(*a b (m + n) == (a + b) m n*)

addendum (3.10.2020)

The magic answer by hand $$a^2 (m - b) (b - n) -b^2 (n - a) (a - m)$$ can easily be simplified to

0==Factor[a^2 (m [Minus] b) (b [Minus] n) -b^2 (n [Minus] a) (a [Minus] m)] // FullSimplify 
(*(a - b) (-a m n - b m n + a b (m + n))==0*)

The second part is equivalent to the condition found by Weierstrass-substitution!

Only provide another way similar to Eliminate. Not so beautiful.

Here we add extra condition.

a ∈ Reals, b ∈ Reals, x ∈ Reals, y ∈ Reals, m ∈ Reals, n ∈ Reals, a != b, a != m, b != n, m != b, n != a

Clear["`*"];
sys = {a*Tan[x] == b*Tan[y], a*Sin[x]^2 + b*Cos[x]^2 == m, 
   b*Sin[y]^2 + a*Cos[y]^2 == n, a ∈ Reals, 
   b ∈ Reals, x ∈ Reals, y ∈ Reals, 
   m ∈ Reals, n ∈ Reals, a != b, a != m, b != n, 
   m != b, n != a};
reg = ImplicitRegion[sys , {a, b, x, y, m, n}];
Resolve[Exists[{x, y}, Element[{a, b, x, y, m, n}, reg]], 
  Reals] // Simplify