P148. Show that there is no real constant c > 0 such that (\cos\sqrt{x+c}=\cos\sqrt{x}) for all real numbers (x\ge 0).
Solution:

If the given equation holds for some constant c>0 then,

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