If `f(x) = cos(log x)` then `f(x)f(y)-1/2[f(x/y)+f(xy)]` has the value

Options

A. 1
B. 0
C. -1
D. none of these

Answer

B

Explanation

we have ,
`f(x)=cos (log(x))`
`:. F((x)/(y))=cos{log((x)/(y))=cos(logx-logy)`
and , `f(xy)=cos{log (xy)}=cos(logx+logy)`
Now, `f((x)/(y))+f(xy)=cos(logx-logy)+cos(logx+logy)`
`implies f((x)/(y))=f(xy)=2 cos (logx) cos (logy)`
`implies (1)/(2){f((a)/(y))+f(xy)}=cos (logx) cos (logy)=f(x)f(y)`
`implies f(x)f(y)-(1)/(2){f((x)/(y))+f(xy)}=0`