Find the Value of the Given Expression
If
\[ f(x)=\cos(\log_e x) \]
then
\[ f\left(\frac1x\right) f\left(\frac1y\right) – \frac12 \left\{ f(xy)+f\left(\frac{x}{y}\right) \right\} \]
is equal to
(a) \(\cos(x-y)\)
(b) \(\log(\cos(x-y))\)
(c) \(1\)
(d) \(\cos(x+y)\)
\[ f\left(\frac1x\right) = \cos\left(\log\frac1x\right) = \cos(-\log x) = \cos(\log x) \]
Similarly,
\[ f\left(\frac1y\right)=\cos(\log y) \]
Using
\[ 2\cos A\cos B = \cos(A+B)+\cos(A-B) \]
\[ 2f\left(\frac1x\right) f\left(\frac1y\right) = \cos(\log xy) + \cos\left(\log\frac{x}{y}\right) \]
\[ = f(xy)+f\left(\frac{x}{y}\right) \]
Therefore,
\[ f\left(\frac1x\right) f\left(\frac1y\right) – \frac12 \left\{ f(xy)+f\left(\frac{x}{y}\right) \right\} =0 \]
\(0\) is not among the options.
\[ \boxed{\text{Correct Answer: None of these}} \]