Difficult complex number question

The question:

Let and w = (1 + i). Find Arg(z) and Arg(w) and hence Arg(zw). Evaluate zw and hence show that . Find a similar expression for

I found:

Arg(z) =

Arg(w) =

Arg(zw) =

zw =

The last bit of the question is giving me trouble. I tried using de Moivre's theorem as follows:

This gives me the argument I want. But I'm unsure of how to solve this. Changing it to sin/cos form (not sure of the proper name for it) seems to be the way to go, but I'm still stumped:

Any assistance would be great!