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