Find the Value of the Given Expression
If \( f(x)=\cos(\log x) \), then find the value of
\[ f(x)f(4)-\frac12\left\{ f\left(\frac{x}{4}\right)+f(4x) \right\} \]
(a) \(1\)
(b) \(-1\)
(c) \(0\)
(d) \(\pm1\)
Using
\[ 2\cos A\cos B=\cos(A-B)+\cos(A+B) \]
\[ 2f(x)f(4) = 2\cos(\log x)\cos(\log 4) \]
\[ = \cos(\log x-\log 4) + \cos(\log x+\log 4) \]
\[ = f\left(\frac{x}{4}\right)+f(4x) \]
Therefore,
\[ f(x)f(4)-\frac12\left\{ f\left(\frac{x}{4}\right)+f(4x) \right\}=0 \]
\[ \boxed{\text{Correct Answer: (c)}} \]