Consider any particle executing SHM with origin as its equilibrium position under the influence of restoring force F=kx , where k is the force constant and x is the displacement of particle from the equilibrium position.Now since F= -kx is the restoring force and from Newtons law of motion force is give as F=ma , where m is the mass of the particle moving with acceleration a. Thus acceleration of the particle is 
     a= F/m

       = -kx/m
but we know that acceleration a=dv/dt=d²x/dt²
⇒         d²x/dt²  = -kx/m   ......(1)
This equation 1 is the equation of motion of SHM.If we choose a constant φ=√(k/m) then equation (1) would become
  d²x/dt²  = -Φ²x  ........(2)

This equation is a differential equation which says that displacement x must be a function of time such that when its second derivative is calculated the result must be negative constant multiplied by the original function.

Sine and cosine functions are the functions satisfying above requirement and are listed as follows :
 x = A Sin ωt  .............(3a)

 x = A Cos ωt ...............(3b)

 x = A Cos (ωt+φ)  .......(3c)

each one of equation 3a, 3b and 3c can be submitted on the left hand side of equation 2 and can then be solved for varification.

Convinently we choose equation 3c i.e., cosine form for representing displacement of particle at any time t from equilibrium position. Thus,

 x = A Cos (ωt+φ)  .......(4)