If $f(x)=\cos \log x$ then $f(x).f(y)-\large\frac{1}{2}$$\bigg(f(\large\frac{x}{y}$)$+f(xy)\bigg)= ?$

Question

$\begin{array}{1 1} \large\frac{-1}{2} \\ \frac{1}{2} \\ -2 \\ 0\end{array} $