Representing the signal $ u(t)=A_c \cos (2 \pi f_c t + \beta \cos 2 \pi f_m t) $ in terms of Bessel function

Question

I want to represent the signal $ u(t)=A_c \cos (2 \pi f_c t + \beta \cos 2 \pi f_m t) $ in terms of Bessel function. I have done the exercise. I want someone to check whether what I have done is correct or not. Thanks for helping.