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If $f(x)=coslogx$ then $f(x).f(y)-\large\frac{1}{2}$$\bigg(f(\large\frac{x}{y}$)$+f(xy)\bigg)= ?$

$\begin{array}{1 1} \large\frac{-1}{2} \\ \frac{1}{2} \\ -2 \\ 0\end{array} $

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Toolbox:
  • $logxy=logx+logy$
  • $log\large\frac{x}{y}$=$logx-logy$
  • $cos(A+B)=cosAcosB-sinAsinB$
  • $cos(A-B)=cosAcosB+sinAsinB$
$f(x).f(y)=coslogx.coslogy$
$f(\large\frac{x}{y})$ $=coslog\large\frac{x}{y}$
$=cos(logx-logy)$
$f(xy)=coslogxy=cos(logx+logy)$
$f(\large\frac{x}{y})$ $+f(xy)=$
$(coslogx.coslogy+sinlogx.sinlogy)+$
$(coslogx.coslogy-sinlogx.sinlogy)$
$=2coslogx.coslogy$
$\large\frac{1}{2}$$\bigg(f(\large\frac{x}{y})$ $+f(xy)\bigg)=coslogx.coslogy$
$f(x).f(y)$$-\large\frac{1}{2}$$\bigg(f(\large\frac{x}{y})$$+f(xy)\bigg)=0$
answered May 18, 2013 by rvidyagovindarajan_1
 

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