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

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Q. If $ f(x)=\cos (\log x), $ then $ f(x)f(y)-\frac{1}{2}\left[ f\left( \frac{x}{y} \right)+f(xy) \right] $ has the value

JamiaJamia 2006

Solution:

Given $ f(x)=\cos (\log x) $ $ \therefore $ $ f(x).f(y)-\frac{1}{2}\left[ f\left( \frac{x}{y} \right)+f(xy) \right] $ $ =\cos (\log x).\cos (\log y) $ $ -\frac{1}{2}\left[ \cos \log \left( \frac{x}{y} \right)+\cos \log (xy) \right] $ $ =\cos (\log x)\cos (\log y)-\frac{1}{2}2[\cos (\log x) $ $ \times \cos (\log y)] $ $ =\cos (\log x)\cos (\log y) $ $ -\cos (\log x)\cos (\log y) $ $ =0 $ $ \therefore $ $ f(x).f(y)-\frac{1}{2}\left[ f\left( \frac{x}{y} \right)+f(xy) \right]=0 $