Question

If $f\left(x\right)=\sin \left(\log x\right),$ then show that $f\left(xy\right)+f\left(\frac{x}{y}\right)-2f\left(x\right)\cos \left(\log y\right)=0$

Answer 1

Consider, $f\left(xy\right)+f\left(\frac{x}{y}\right)-2f\left(x\right)\cos \left(\log y\right)$

$=\sin \left(\log xy\right)+\sin \left(\log \frac{x}{y}\right)-2\sin \left(\log x\right)\cos \left(\log y\right)$

$=\sin (\log x+\log y)+\sin (\log x-\log y)-2\sin \left(\log x\right)\cos \left(\log y\right)$

Applying $\sin C+\sin D=2\sin \frac{(C+D)}{2}\cos \frac{(C-D)}{2}$, we get

$=2\sin \log x\cos \left(\log y\right)-2\sin \log x\cos \left(\log y\right)$

$=0$