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)

Subtracting equations (ii) from (i), we get : 

$$cos(x+y) - cos(x-y) = -2sinxsiny$$ ------------(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)

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

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

=> Ans - (C)