If cosC + cosD = x, then value of x is

We know that, $$cos(x+y) = cosxcosy - sinxsiny$$ --------(i)

and $$cos(x-y) = cosxcosy + sinxsiny$$ ------------(ii)

Adding equations (i) and (ii), we get : 

$$cos(x+y) + cos(x-y) = 2cosxcosy$$ ------------(iii)

Let $$x + y = C$$ and $$x - y = D$$

=> $$x = \frac{C+D}{2}$$

and $$y = \frac{C-D}{2}$$

Substituting above values in equation (iii)

=> $$cosC + cosD = 2cos(\frac{C+D}{2})cos(\frac{C-D}{2})$$

=> Ans - (A)