find a different form for trigonometric expression

Hi, I need to prove that:

x[t]=cos(ω0*t)+ cos( ωo*t + Δω*t)

can be transform into the form:

x[t]=A(t)*cos[ωo*t + θ(t)]

where A(t) and θ(t) are function of Δω.

I have the solution but I cannot find out the way to solve it

A(t)=2|cos(Δω*t)|

and

θ(t)= ArcTan[sin(Δω*t)/(1+cos(Δω*t))]

I have started by using the trigon identity cos(a+b).

I first factor cos[ωo*t] to have 1+cos(Δω*t) then I factor 1+cos(Δω*t) to have the expression under the Arctan.

At one point I get:

x[t]= (1+cosΔω*t)*[cos[ωo*t]-sin(ωo*t)*{sin(Δω*t)/(1+cosΔω*t)]}]

from here I can not figure out how to fin A(t) and θ(t).

please can someone help me to finsh the computation?

thank you

B