John says that the value of the function cos[ω(t + t) + ϕ], obtained one period t after time t, is greater than cos(ωt + ϕ) by 2π. larry says that it is greater by the addition of 1.00 to cos(ωt + ϕ). which one, if either, is correct?

No one is right

Explanation:

John Case:

The function is defined between -1 and 1, So it is not possible obtain a value greater.  

In addition, if you  move the function cosine a T Value, and T is the Period,  the function take the same value due to the cosine is a periodic function.

Larry case:

Is you have , the domain of this is [0,2].

it is equivalent to adding 1 to the domain of the , and its mean that the function , in general, is not greater than .