A) \[\cos (x-y)\]
B) \[\log (x-y)\]
C) \[\cos (x+y)\]
D) None of these
Correct Answer: D
Solution :
Now, \[f\left( \frac{1}{x} \right)f\left( \frac{1}{y} \right)-\frac{1}{2}\left[ f\left( \frac{x}{y} \right)+f(xy) \right]\] \[=\cos \left( \log \frac{1}{x} \right)\cos \left( \log \frac{1}{y} \right)\] \[-\frac{1}{2}\left[ \cos \left( \log \frac{x}{y} \right)+\cos [\log (xy)] \right]\] \[=\cos (-\log x)\cos (-\log y)\] \[-\frac{1}{2}[\cos (\log x-\log y)+\cos (\log x+\log y)]\] \[=\cos (\log x)\cos (\log y)\] \[-\frac{1}{2}[2\cos (\log x)\cos (\log y)]\] \[=\cos (\log x)\cos (\log y)\] \[-\cos (\log x)\cos (\log y)=0\]
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