If $$f(x)=$$$$cos$$⁡loge⁡x, then $$f(x)f(y)$$−$$12fxy+f(xy)has$$ value

Correct option is 4None of these

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$If f (x) = \cos (\log_{e} x), then f (x) f (y) - \frac{1}{2} [f (\frac{x}{y}) + f (x y)] h a s v a l u e$

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detailed solution

Correct option is D

fx=cos logx ⇒ fxfy-12fxy+fxy =cos logx cos logy -12coslogx -logy+coslogx+logy=coslogx coslogy-122 coslogx coslogy since COSA+B+COSA-B=2 COS A COS B=0

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If f(x)=cos⁡loge⁡x, then f(x)f(y)−12fxy+f(xy)has value

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$If f (x) = \cos (\log_{e} x), then f (x) f (y) - \frac{1}{2} [f (\frac{x}{y}) + f (x y)] h a s v a l u e$