Question

If $f\left(x\right)=\cos \left[{\pi }^2\right]x+\cos [-{\pi }^2]x$, where $\left[x\right]$ denotes the greatest integer less than or equal to x, then write the value of $f\left(\pi \right)$.

Answer 1

$f\left(x\right)=\cos \left[{\pi }^2\right]x+\cos [-{\pi }^2]x$
${\pi }^2\approx 9.8596$
So, we have $\left[{\pi }^2\right]=9\text{and}[-{\pi }^2]=-10$
$f\left(x\right)=\cos 9x+\cos (-10)x$
$=\cos 9x+\cos 10x$
$f\left(\pi \right)=\cos 9\pi +\cos 10\pi$
$\Rightarrow -1+1$
Therefore, $f\left(\pi \right)=0$