Sinusoids represented using Euler's formula notation, confusion
What you are actually trying to find out here, is for what operators \$H\$ you have the following property:

$$\mathcal{R}\left\{H\left\{f(x,t)\right\}\right\} = H\left\{\mathcal{R}\left\{f(x,t)\right\}\right\}$$

Or otherwise, which operators \$H\$ are commutative with taking the real part. Any operator that does not have this property (like squaring) will give rise to the issues you are addressing.

So using the exponential notation is only valid if the problem never requires you to use an operator that doesn't satisfy that property. Most problems are linear though, and all linear operations are commutative with taking the real part.


  Can someone clear up my confusion? Since, the above is false, how can we simply work with \$e^{j(\omega t - kx)}\$


So to answer your question: you can't in your example.