In problem 6 of the Exponential Project, the formula for Re[exp(z)] is correct for pure imaginary z=it, which is the case discussed in Rudin. For general z, the bar in the second term in the numerator should be over the z rather than where it landed in front of it.
Keep in mind that the goal of the projects and lectures we are working on now is to continue using the material we learned before in new contexts so that you can consolidate it and stay in practice while demonstrating to yourself and others that you are now equipped to master some advanced mathematics. That does not mean you have to master all of it. As I have said, you are not responsible for doing or understanding every detail the way you are with the Notes. And the CAs and I are happy to give lots of hints with the parts you want to work through. Happy 2000, --DLG