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Two eforts in Mathematica 5.2 in order
to simplify o below

o = 2*Cos[Pi/48]*(1 - Cos[Pi/48]^2)^(1/2) + 2*Cos[Pi/48]^2 - 1;

First leaving Mathematica do the job.

Timing[ToRadicals[RootReduce[(FullSimplify[#1, ComplexityFunction ->
(Count[{#1}, _Cos, Infinity] & )] & )[o]]]]
{43.031*Second, Sqrt[(1/2)*(2 + Sqrt[2 - Sqrt[3]])]}

Then help a bit Mathematica.
I couldn't find a built in command to simplify 1 - Cos[Pi/48]^2 to
Sin[Pi/48]^2. That's why the adding of a rule.

Timing[ToRadicals[RootReduce[Simplify[o /. 1 - Cos[Pi/48]^2 ->
{0.25*Second, Sqrt[(1/2)*(2 + Sqrt[2 - Sqrt[3]])]}

Any other suggestions will be highly appreciated.


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