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

Question

\[\begin {array} {1 1} (a)\;\bigg\{(x,y):\cos \bigg[\frac{x-y}{2}\bigg]=\frac{1}{2}\bigg\} & \quad (b)\;\bigg\{(x,y):\sin \bigg[\frac{x-y}{2}\bigg]=\frac{1}{2}\bigg\}\\ (c)\;\bigg\{(x,y):\cos (x-y)=\frac{1}{2}\bigg\} & \quad (d)\;empty\;set \end {array}\]