From  sin(x+y) =  = ,  it follows that

    x + y = 45°    OR   x + y = 135°.



From  cos(2x) = - ,  it follows that  

    2x = 120°      OR   2x = 240°,   i.e.

     x = 60°       OR    x = 120°.


Therefore, you need to solve 4 (four) systems of equations


     1)  x + y =  45   (mod 360)
         x     =  60   (mod 360)


     2)  x + y = 135   (mod 360)
         x     =  60   (mod 360)


     3)  x + y =  45   (mod 360)
         x     = 120   (mod 360)


     4)  x + y = 135   (mod 360)
         x     = 120   (mod 360)


Solution of each and every system  1)  to  4)  is an elementary task.


The solution to the original problem will be the union of the solutions of partial problems  1) - 4).