Solution:

Given, f(x) = cos(logex)

To find: f(1/x)f(1/y) -1/2{f(xy) f(x/y)}

Let's calculate each term separately.

1. f(1/x) = cos(loge(1/x)) = cos(-logex) = cos(logex) = f(x)

2. f(1/y) = cos(loge(1/y)) = cos(-logey) = cos(logey) = f(y)

3. f(xy) = cos(loge(xy)) = cos(logex + logey) = cos(logex)cos(logey) - sin(logex)sin(logey)

4. f(x/y) = cos(loge(x/y)) = cos(logex - logey) = cos(logex)cos(logey) + sin(logex)sin(logey)

Substituting the above values in the given expression, we get:

f(1/x)f(1/y) -1/2{f(xy) f(x/y)}

= f(x) f(y) - 1/2{[f(x)f(y) - f(x)f(y)]}

= f(x)f(y) - 1/2{0}

= f(x)f(y)

= cos(logex)cos(logey)

= cos(logexy)

Therefore, the answer is option (a) cos(x-y).

Explanation:

We have used the trigonometric identity cos(a+b) = cos(a)cos(b) - sin(a)sin(b) and cos(a-b) = cos(a)cos(b) + sin(a)sin(b) to simplify the expression.

By substituting the values of f(1/x), f(1/y), f(xy), and f(x/y) in the given expression, we have simplified it to f(x)f(y). Finally, we have used the property of logarithms to simplify cos(logexy) to cos(x-y).

Hence, the correct answer is option (a) cos(x-y).

Given ; f(x)= cos(loge x)= cos (log x) {since e is the base of log x}.