Question

Given that x=Acos(wt) is a sinusoidal function of time, show that v(velocity) and a(acceleration) are also sinusoidal functions of time. (HINT: Use v=square root [(k/m)(A^2-x^2)] and a=(-k/m)X

The hint tells you what to do: put Acoswt for X.

A second way which is easier is in calculus:
v= dx/dt
a= dv/dt= d"x/dt"

That is the most simple way.