Thread: Geometry Proof based on Trigonometry and Cosine Law

Hello everybody. Because this question is more challenging and combines geometry and tirgonometry, I posted it here. I hope this is ok.

I'm not sure if this diagram below is wrong or if I've forgotten geometry. But here are the issues I'm trying to deal with that I've written on the actual solution.

The black is the actual question. The purple is the proposed solution that I'm having doubts about. Thanks a lot.