UPSEE Solved Model Paper 7

Given:
f (x) = cos (log x)
We have to find the value f (x) . f (y) -

1

2

[f(x−y)+f(xy)]
Now, f (x) . f (y) -

1

2

[f(x−y)+f(xy)]
= cos (log x) . cos (log y) -

1

2

[cos(log‌

x

y

)+cos(log‌x‌y)]
= cos (log x) . cos (log y) -

1

2

[cos (log x - log y) + cos (log x + log y)]
[Using properties of logarithmic function 1 and properties of logarithmic function 2]
= cos (log x) cos (log y) -

1

2

[cos (log x) cos (log y) + sin (log x) sin (log y) + cos (log x) cos (log y) - sin (log x) sin (log y)]
[Using transformation formula 1]
= cos (log x) cos (log y) -

1

2

[2cos (log x) cos (log y)]
= cos (log x) cos (log y) -

1

2

[2cos (log x) cos (log y)]
= cos (log x) cos (log y) - cos (log x) cos (log y) = 0
Hence, option 'D' is correct.