Proving the Addition Formula for Complex Functions

[kasse]

How can I show that

$$Aexpi(\omega t - kx + \phi_{1}) + Aexpi(\omega t + kx + \phi_{2}) = 2Aexpi(\omega t + \frac{\phi_{1} + \phi_{2}}{2})cos(kx - \frac{\phi_{1} - \phi_{2}}{2})$$? My book states this without proof, as if it was obvious.

 

[tiny-tim]

Hi kasse!

Hint: e^{iA + B} = e^iA + e^iB, so divide by the first half of the righthand side, and use e^iC + e^-iC = 2cosC.

 

[kasse]

Danke schön

 

[HallsofIvy]

Hi kasse!

Hint: e^{iA + B} = e^iA + e^iB, so divide by the first half of the righthand side, and use e^iC + e^-iC = 2cosC.

Oh, dear, oh, dear, oh, dear, tiny-tim! e^{iA + B} = (e^iA)(e^iB).

 

[tiny-tim]

oops!

Oh, dear, oh, dear, oh, dear, tiny-tim! e^{iA + B} = (e^iA)(e^iB).

Geman for oops!

 

[kasse]

e^{iA + B} = (e^iA)(e^B)...?