If this claim is not true, then there exists a constant positive real number $T$ such that for every real number $x$, the relation below always hold. \begin{equation} \cos\sqrt{x+T} = \cos\sqrt{x} \end{equation} Setting $x=0$ leads to $$\cos\sqrt{T}=\cos 0 = 1\implies \sqrt{T}=2k\pi\qquad(k\in\mathbb Z)$$ Next, setting $x=T$ and also noting $\cos\sqrt[]{T}=1$ yield $$\cos\sqrt[]{2T}=\cos\sqrt[]{T}=1\implies \sqrt[]{2T}=2m\pi\qquad(m\in\mathbb{Z})$$ $$\therefore\quad \sqrt[]{T}=\frac{1}{\sqrt[]{2}}\cdot(2m\pi)=2k\pi\implies \sqrt[]{2}=\frac{m}{k}$$ However, this is impossible because $\sqrt{2}$ is irrational, but the right side is rational.