The proof is below.
Work Step by Step
We find: $=Acos(kx-\omega t )+ Acos(kx-\omega t +\phi/2)$ Using trigonometry, we can simplify this to: $=2A cos(kx-\omega t +\phi/2)$. Thus, from the inside of the cosine function, we see that the wave is a simple harmonic wave. The amplitude is 2A, but the book wants this in terms of the phase constant, so we find that the amplitude is $2Acos(\phi/2)$