Question

If f(x)=cos(logx) then f(x)f(y)-1/2[f(x/y)+f(x)] is equal to

Answer 1

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

Then

$f(x/y)=\cos (\log (x/y))=\cos (\log x-\log y)\\ f(xy)=\cos (\log (xy))=\cos (\log x+\log y)$

Now,

$f(x/y)+ f(xy)=\cos (\log x-\log y)+\cos (\log x+\log y)\\ f(x/y)+ f(xy)=2\cos (\log x)\cos (\log y)\\ f(x/y)+ f(xy)=2f(x)f(y)\\ f(x/y)+ f(xy)-2f(x)f(y)=0\\ f(x)f(y)-\frac{1}{2}[f(x/y)+ f(xy)]=0$

So the given expression is equal to 0.