How can I show that
\(Aexpi(\omega t - kx + \phi_{1}) + Aexpi(\omega t + kx + \phi_{2}) = 2Aexpi(\omega t + \frac{\phi_{1} + \phi_{2}}{2})cos(kx - \frac{\phi_{1} - \phi_{2}}{2})\)?

 

How can I show that
\(Aexpi(\omega t - kx + \phi_{1}) + Aexpi(\omega t + kx + \phi_{2}) = 2Aexpi(\omega t + \frac{\phi_{1} + \phi_{2}}{2})cos(kx - \frac{\phi_{1} - \phi_{2}}{2})\)?

One way is to factor out \(2Aexpi(\omega t + \frac{\phi_{1} + \phi_{2}}{2})\) from the left hand side.

You will then be left with a much easier problem, which is to show that

\({{expi( - kx + \phi_{1}-\frac{\phi_{1} + \phi_{2}}{2}) + expi( kx + \phi_{2}-\frac{\phi_{1} + \phi_{2}}{2})}\over{2}} = cos(kx - \frac{\phi_{1} - \phi_{2}}{2})\)