You can NEVER solve for those variables, at least as the equations are written.
First of all, z never even appears. But since you never try to solve for it, who cares?
But look at x + y. They always appear tegether, in EXACTLY the same way. In fact, we can transform the problem in a very simple way, by replacing (x+y) with z.
Now your problem reduces to:
eq1 = -sin(z)+cos(z-1.6)== 0
eq2 = cos(z)+cos(z-3.2)== 0
There is no more and no less information content in the problem. Surely you can agree, since x+y ALWAYS appear in exactly the same place, and always together.
pi/4 + 4/5
Solving eq1 for z, we find a simple solution. Solving eq2 for z, however, we get a completely different value for z. It is not really imaginary when we resolve it into a simpler numeric form. But see that it is not the first value we got for z.
So nothing in the universe will ever allow a solution to both equations for z, and therefore solving for x and y is just as impossible. It is very easy to write equations that have no solution. Surprise! You just did it.