In Simple Harmonic Motion, the period $T$ of an oscillation, is said to be independent of the amplitude $A$ of an oscillation, but why is that so?
Attempting to derive from the equations of Simple Harmonic Motion, doesn't seem to get me anywhere :
$$x(t) = A\cos{(\omega t)}$$ $$\implies x(t) = A\cos{(\frac{2\pi}{T} t)}$$
But it is unclear to me how show the independance of $T$ from $A$ from the above equation, or even if it can be shown through a derivation here.