I haven't found a way to get `DSolve` to crack it, so turn to `NDSolve`

    a=1;d=2;
    sol={y[t],x[t]}/.NDSolve[{
      I y'[t] == Cos[a t] Exp[-I d t] x[t], I x'[t] == Cos[a t] Exp[I d t] y[t],
      y[0]==1,x[0]==1}, {y[t], x[t]}, {t,0,2}][[1]];
    Plot[{Re[sol[[1]]],Im[sol[[1]]],Re[sol[[2]]],Im[sol[[2]]]},{t,0,2}]

which instantly gives you plots of the real and complex components of x and y.

Some people think everyone should use `NDSolveValue` and some think `NDSolve`,  some people use `{x,y}` and some use `{x[t],y[t]}`. There are several different ways of doing almost anything in Mathematica. Pick a method that you can remember and use without making too many mistakes.