Proof that $ \sin \sqrt{x}$ is not periodic function using contradiction method?

Question

Proof that $ \sin \sqrt{x}$ is not periodic function using contradiction method ?

My approach For periodic $$f(x+T)=f(x)$$ Let $x=0$ $$\sin \sqrt{(T+0)}=\sin\sqrt{0}$$ $$\sqrt{ T} =n\pi$$ Let $x= T$ $$\sin \sqrt{2T}=\sin\sqrt{T}$$

After this I can't able to solve

Answer 1

Hint: Use that $$\sin(x)-\sin(y)=2 \sin \left(\frac{x}{2}-\frac{y}{2}\right) \cos \left(\frac{x}{2}+\frac{y}{2}\right)$$