Concept of log and trigonometric

Question

If !!f(x)= cos(logx)!!. Then !!f(x//y)+f(xy)!!

(a) !!2f(x)-2f(y)!!

(b) !!f(x)- f(y)!!

(c) !!f(x)* f(y)!!

(d) !!2f(x)* f(y)!!

Solution

!!f(x)= cos(logx)!!

!!f(x//y)+f(xy)= cos(log(x//y))+ cos(log(xy))!!

!!=> cos(logx-logy)+ cos(logx+logy)...(1)!!

We know that: !!cos(A+B)+ cos(A-B)= 2cosAcosB!!

From equation (1), we will get

!!f(x//y)+f(xy)= 2cos(logx)cos(logy)!!

!!=> f(x//y)+f(xy)= 2 f(x)f(y)!!

Therefore, option (d) is correct.