rwg@sdf.lonestar.org wrote:
BCCee Dick Askey replied: Here are two similar identities. tan(3pi/11) + 4sin(2pi/11) = 11^(1/2) sin(2pi/7) + sin(4pi/7) + sin(8pi/7) = (1/2)*7^(1/2). These are in Chapter 7 of Hobson's "A Treatise on Plane and Advanced Trigonometry". In case you do not know this book, it has a number of gems. Dover has it back in print. Dick
Mma 7.0 FullSimplify does these, too. Maybe it just guesses them numerically? No, it has a ferocious algebraic number simplifier:
In[39]:= ToRadicals[Tan[3*Pi/11] + 4*Sin[2*Pi/11]] Out[39]= 5/11 7/22 4/11 I (1 + (-1) ) -2 (-1) (-1 + (-1) ) + ---------------- 5/11 -1 + (-1)
In[40]:= RootReduce[%]
Out[40]= Sqrt[11] --rwg
Gosper is mentioned prominently and favorably in the latest issue of the American Mathematical Monthly.