Grievous is probably the worst character in the movie, the cough cough cough is unexplained-we're supposed to rely on the now non-canon DVD movie Clone Wars (not the recent Clone Wars that is canon) to find out Mace Windu wacked Grievous one in the chest for him to start coughing. That was fine and dandy in 2005 when a lot of people had watched the Clone Wars anime, but that is not fine in 2016 when no one has a clue that thing even existed and Grievous just coughs coughs coughs his way to failure.
The cough that Grievous has isn't crucial to the plot. You don't really need an explanation.
It's also stupid that Obi-Wan goes into Episode III with NO idea that Palpatine is the sith lord they are looking for, despite being told in the third quarter of Attack of the Clones that the Sith were leading the Galactic Republic & Senate.
It's easy to say Obi-Wan is stupid when you're privy to information that he isn't. From his point of view, Dooku may have been trying to create mistrust between the Jedi and the Senate.
"Have to say, Jimmy - you turned into a real impressive fighter. If I was ol' Mengsk, I'd be runnin' scared from you, too."
"You really mean that, Tychus?"
"Nah. I was just kiddin'."
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