FM and phase/frequency deviation
We know that Ufm(t) = Acos(ωt + βsinωt)


It's better to distinguish the two omegas you have in the above equation as 1 and 2 (subscripts).

Back to the question....

To find the max frequency deviation it can be assumed that this will be when the two individual sine waves are both hitting maximum or minimum amplitude simulataneously. Unfortunately, the two sine terms are locked in phase with one double the frequency of the other so you need to differentiate the sum of the two sine terms and equate to zero. This should be fairly straightforward.