If the given equation holds for some constant c>0 then,
f(x) = $latex (\cos\sqrt{x}-\cos\sqrt{x+c}=0)$ for all $latex (x\ge 0)$
$latex (\Rightarrow 2\sin\frac{\sqrt{x+c}+\sqrt{x}}{2}\sin\frac{\sqrt{x+c}-\sqrt{x}}{2}=0)$
Putting x=0, we note
$latex (\Rightarrow\sin^2\frac{\sqrt{c}}{2}=0)$
As $latex (c\not=0)$
$latex (\sqrt{c}=2n\pi)$
$latex (\Rightarrow c=4n^2\pi^2)$
We put n=1 and x=$latex (\frac{\pi}{2})$ to note that f(x) is not zero.
Hence no c>0 allows f(x) =0 for all $latex (x\ge 0)$. (proved)
