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Question
If f(x) = cos (log x), then the value of f(x) f(y) −\[\frac{1}{2}\left\{ f\left( \frac{x}{y} \right) + f\left( xy \right) \right\}\] is
Options
(a) −1
(b) 1/2
(c) −2
(d) None of these
Solution
(d) None of these
Given:
\[ \Rightarrow f\left( \frac{x}{y} \right) + f\left( xy \right) = 2\cos\left( \log x \right)\cos\left( \log y \right)\]
\[ \Rightarrow \frac{1}{2}\left[ f\left( \frac{x}{y} \right) + f\left( xy \right) \right] = \cos\left( \log x \right)\cos\left( \log y \right)\] \[\Rightarrow f\left( x \right)f\left( y \right) - \frac{1}{2}\left\{ f\left( xy \right) + f\left( \frac{x}{y} \right) \right\} = \cos\left( \log x \right)\cos\left( \log y \right) - \cos\left( \log x \right)\cos\left( \log y \right) = 0\]
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