I don't know why Wolfram Alpha cannot handle this system. Mathematica finds the real solution this way:
ContourPlot[{Cos[x]*Cos[y] + 2*x == 0,
-Sin[x]*Sin[y] + 2*y == 0},
{x, -4, 4}, {y, -4, 4}]
Reduce[Cos[x]*Cos[y] + 2*x == 0 &&
-Sin[x]*Sin[y] + 2*y == 0 &&
Abs[x] < 2 && Abs[y] < 2,
{x, y}, Reals]
There are complex solutions too:
Reduce[Cos[x]*Cos[y] + 2*x == 0 &&
-Sin[x]*Sin[y] + 2*y == 0 &&
Abs[x] < 2 && Abs[y] < 2,
{x, y}]