If $f(x)=\cos(\log x)$, then find the value of $f(x)×f(4)-[\frac{1}{2}]×[f(\frac{x}{4})+f(4x)]$.

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$f(x)=\cos(\log x)$

Now let $y=f(x)×f(4)-[\frac{1}{2}]×[f(\frac{x}{4})+f(4x)]$

$y=\cos(\log x)×\cos(\log 4)-[\frac{1}{2}]×[\cos \log(\frac{x}{4})+\cos(\log 4x)]$

$y=\cos(\log x)\cos(\log 4)-[\frac{1}{2}]×[\cos (\log x-\log 4)+\cos (\log x+\log 4)]$

$y=\cos(\log x)\cos(\log 4)-[\frac{1}{2}]×[2\cos(\log x)\cos (\log 4)]$

$y = 0$