How could Count Dooku lose against Anakin Skywalker in a lightsaber battle?
At Wookieepedia I found the following:
Dooku was a respected instructor in the Jedi Temple and one of the most renowned swordsmen in the galaxy. Only Masters Yoda and Mace Windu were considered his equals.
When it comes to lightsaber combat mastery one finds regularly these three names, sometimes alongside Darth Sidious, but not that of Anakin Skywalker or Darth Vader. Examples:
Mace Windu was a formidable warrior and generally considered one of the greatest swordsmen ever produced by the Jedi Order [...]. Even without his unique fighting style, it is said that only his one-time friend, Dooku, and the venerable Grand Master Yoda could outspar him.
Yoda was a master of lightsaber combat, one of the greatest duelists of all time. It was said that only Mace Windu, Count Dooku and Palpatine were able to fight with him on equal terms.
So how could Anakin defeat Count Dooku so easily at the beginning of The Revenge of the Sith?
This Question contains some good points but I guess it can't be only a matter of fighting styles as Dooku's "Makashi" is a special dueling style and he had good knowledge of the other styles too.
Maybe Dooku's age played a role but then again when the force is your ally this shouldn't matter too much (see Yoda).
According to the novel, Sidious had told Dooku to hold back, because he wanted to use Anakin to murder Obi-Wan. (Which also partly explains Dooku's shocked face when Sidious tells Anakin to kill Dooku. While he might have expected a Sith master to stand back and let the strongest man win, he would not have expected his master to change the plan on him in that way.)
Also, Sidious was there and he was watching. He was also quite capable of using the Force in whatever small ways he wanted to if it would influence the outcome by either helping Anakin or by hindering Dooku.
Also stated in the book, Dooku's pride got in his way. Anakin and Obi-Wan's opening stances were from different "styles" than they proceeded to fight in. Dooku was put on an unbalanced defensive by preparing for their fighting style based on these opening stances, and then not fighting as he expected them to.
But didn't he only hold back at first and "powered up" after noticing that Obi-Wan and Anakin held back as well to ensure he could kill Obi-Wan?
I'm not clear on what you mean, but if he was told to hold back by Sidious, he'd have to hold back. Not hold back, then get better.
I'm sorry, if I was a bit unclear. If I remember it correctly the Count had at first a very easy time fighting Anakin and Obi-Wan, because the both of them held back at first. Then the two jedi changed tactics revealing that they have gotten much better in lightsaber combat. Thus they surprised the Count and he got into trouble. But Dooku was able recover and continued the fight with (I presume) full strength, so that he was able to take out Obi-Wan.
You're presuming full strength, but a powerful Sith Master, his own master, was right there and had told him to take it easy. It's unlikely that he'd be disobeying Sidious when he's right there watching. He may want to look like he's trying real hard, though.
In the novel you find: "[Dooku thought]: These clowns might—just possibly—actually be able to beat him. No sense taking chances; even his Master would agree with that. Lord Sidious could come up with a new plan more easily than a new apprentice." This sounds to me like he decided to take them on with full strength so as not to risk his own life.
Furthermore: "Then Sidious, for some reason, decided to intervene. "Don't fear what you're feeling, Anakin, use it!" he barked in Palpatine's voice. "Call upon your fury. Focus it, and he cannot stand against you. Rage is your weapon. Strike now! Strike! Kill him!" Dooku thought blankly, Kill me? [...] Dooku found himself wondering in bewildered astonishment if Sidious had suddenly lost his mind. [...] and he felt a sickening presentiment that he already knew the answer to that question. Treachery is the way of the Sith." So he knew that his master had betrayed him and his life was at risk.
@Philipp: And that's when he gave in to anger and fear. And we know what that does (even if the Sith use such emotions).
This question always bugged me! However after reading these comments I'm like, so because Anakin gave into his dark side tapping into fear, fury etc etc he became powerful enough to best Dooku. But isn't this what Luke did to Vader in ROTJ? and if so, how come he didn't falter to the dark side? sounds like a major plot hole.
@Jared: Luke and Anakin both gave in to anger at some point, but Luke didn't fall because he stopped himself at the edge - when told to kill Vader, he refused. When Anakin was told to kill Dooku, he... refused less.