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  4. If f(x)= sin log x, then value of f(x y)+f((x/y))-2 f(x) cos log y is equal to

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Q. If $f(x)=\sin \log x$, then value of $f(x y)+f\left(\frac{x}{y}\right)-2 f(x) \cos \log y$ is equal to

KEAMKEAM 2019

Solution:

$f(x y)=\sin \log x y=\sin (\log x+\log y) \,\,\,\ldots( i )$
$f\left(\frac{x}{y}\right)=\sin \log \left(\frac{x}{y}\right)=\sin (\log x-\log y) \,\,\, \ldots($ ii $)$
On adding Eqs. (i) and (ii)
$f(x y)+f(x / y)=2 \sin \log x \cos \log y$
$\therefore f(x y)+f\left(\frac{x}{y}\right)=2 f(x) \cos \log y$
$\Rightarrow f(x y)+f(x / y)-2 f(x) \cos \log y=0$