[math-fun] Machinoid arcsin π formulæ

These must have been done to death. In[526]:= FullSimplify[Sin[-ArcSin[23/361] + 2*ArcSin[11/38]]] Out[526]= 1/2 In[528]:= FullSimplify[Sin[ArcSin[97/961] + 2*ArcSin[13/62]]] Out[528]= 1/2 In[532]:= FullSimplify[Sin[2*ArcSin[11/43] + ArcSin[23/3698]]] Out[532]= 1/2 In[517]:= FullSimplify[Sin[ArcSin[23/98] + 2*ArcSin[1/7]]] Out[517]= 1/2 I.e., ArcSin[97/961] + 2*ArcSin[13/62] = π/6, but Wolfram's nascent artificial intelligence is too stupid to notice. I don't know if this scheme can beat Machin--I only looked at 2 arcsin a + arcsin b. The apparent messiness of the arcsin series vs arctan is illusory if we replace clunky old summation with matrix products. (gosper.org/pathi.pdf). --rwg