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Here is how I am doing it. I am taking the x(t)=A*cos(wt+θ) equation and then the second derivative of it to find acceleration

a(t)=-A*w^2*cos(wt+θ)

F=ma=-kx

a=-kx/m

-A*w^2*cos(wt+θ)=-kx/m

so I am guessing that I can make a few assumptions to solve for the final equation. If I assume that the time t=0, thus we x=A, I can solve

-x*w^2*cos(w(0)+0)=-kx/m

x's cancel out

w^2*cos(0)=k/m

w=(k/m)^(1/2)

Is this a valid way to prove the equation? Also, I am assuming that the original x(t) equation is correct, I do not know how to prove it. Is there a way to prove the original x(T) equation? Finally, I am preparing for the AP Physics C: Mechanics exam, so if I were to solve a problem using Calculus, for those of you familiar with the exam, would the graders allow me to start off a proof of an answer with x(t)=A*cos(wt+θ) as a given?