Question

Let $f\left(x\right)=\cos \left(\log x\right)$ the prove that $f\left(x\right).f\left(y\right)=\frac{1}{2}[f\left(\frac{y}{x}\right)+f(xy\left)\right]$.

Answer 1

Given,

$f\left(x\right)=\log \left(\cos x\right)$.

Now, $\frac{1}{2}[f\left(\frac{y}{x}\right)+f(xy\left)\right]$

$=\frac{1}{2}\left[\cos \right(\log x-\log y)+\cos (\log x+\log y\left)\right]$

$=\cos \left(\log x\right).\cos \left(\log y\right)$ [Using formula of $\cos (A+B)+\cos (A-B)=2\cos A.\cos B$]

$=f\left(x\right).f\left(y\right)$