Question

If $f\left(x\right)=\cos \left(\log x\right)$, than show that $f\left(\frac{1}{x}\right).f\left(\frac{1}{y}\right)-\frac{1}{2}\left[f\left(\frac{x}{y}\right)+\left(xy\right)\right]$

Answer 1

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

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

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

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

Hence proved.