Find the Value of the Given Expression
If \( f(x)=\cos(\log x) \), then find the value of
\[ f(x)f(y)-\frac12\left\{ f\left(\frac{x}{y}\right)+f(xy) \right\} \]
(a) \(-1\)
(b) \(\frac12\)
(c) \(-2\)
(d) none of these
\[ f(x)=\cos(\log x), \qquad f(y)=\cos(\log y) \]
Using
\[ 2\cos A\cos B=\cos(A-B)+\cos(A+B) \]
\[ f(x)f(y) = \frac12\left[ \cos(\log x-\log y) + \cos(\log x+\log y) \right] \]
\[ = \frac12\left[ f\left(\frac{x}{y}\right) + f(xy) \right] \]
Therefore,
\[ f(x)f(y)-\frac12\left\{ f\left(\frac{x}{y}\right)+f(xy) \right\}=0 \]
\[ \boxed{\text{Correct Answer: (d) none of these}} \]