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Mathematica 10 evaluates Simplify[Cos[2 ArcCos[A]]] as Cos[2 ArcCos[A]], instead of the desired 2 A^2 - 1. How do I go about making this reduction?

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Try TrigExpand

TrigExpand[Cos[2 ArcCos[A]]]
-1 + 2 A^2
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    $\begingroup$ Comment: situations like this, where specialized functions such as TrigExpand or TrigToExp are needed, often pop up. They are seemingly counterexamples to one of the Mathematica core principles: "Automate as much as possible." (wolfram.com/language/principles) $\endgroup$ – murray Oct 8 '14 at 14:16

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