Question

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

• A. 0
• B. $\frac{\cos \log x}{x}$
• C. 1
• D. $\frac{1}{2}$
• E. None of these

Answer 1

Answer: (A)

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