The solution of the equations $x+y=\frac{2\pi }{3}$  and $\cos x+\cos y=\frac{3}{2}$ , where $x\text{and}y$  are real is

$\cos x+\cos y=\frac{3}{2}$

$2\cos \left(\frac{x+y}{2}\right).\cos \left(\frac{x-y}{2}\right)=\frac{3}{2}$

$2\cos \frac{\pi }{3}.\cos \left(\frac{x-y}{2}\right)=\frac{3}{2}$

$\therefore \cos \left(\frac{x-y}{2}\right)=\frac{3}{2}$

Solution does not exist