Question

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

Answer 1

Let $f\left(x\right)=\cos \left(\log x\right)$.

Then $f\left(\frac{x}{y}\right)=\cos (\log x-\log y)$ and $f\left(xy\right)=\cos (\log x+\log y)$.

Now

$f\left(x\right).f\left(y\right)-\frac{1}{2}[f\left(\frac{x}{y}\right)+f(xy\left)\right]$

$=\cos \left(\log x\right).\cos \left(\log y\right)-\frac{1}{2}\left[\cos \right(\log x-\log y)+\cos (\log x+\log y\left)\right]$

$=\cos \left(\log x\right).\cos \left(\log y\right)-\cos \left(\log x\right)\cos \left(\log y\right)=0$