JBuddenhagen>Don't know mathematica. Perhaps of interest, wolframalpha also fails to simplify to 0, see http://www.wolframalpha.com/input/?i=arccos%28-%287%2F9%29%29+-+2*arcsec%283... However my old maple 7 outputs 0 for simplify(arccos(-(7/9)) - 2*arcsec(3)); <JB GACK! Instead Malpha expresses the 0 in terms inverse Jacobi elliptic functions! And then asks me to pay for additional computation time. My Macsyma hasn't yet adapted to Mavericks, but I can pretty much predict the outcome. Macsyma's closest (not very) equivalents to the FullSimplify shotgun are rat, ratsimp, and radcan, which are for canonicalizing certain classes of expressions, e.g. rational and algebraic functions, logs, powers, etc. They wouldn't apply here, at least directly. Likewise trigsimp, which isn't for arctrigs. As with Mathematica, trigexpand(cos(%)), or ratsimp thereof, would surely work. But to get 0 directly, I would try radcan(logarc(%)), converting the arctrigs to complex logs. In fact, Mathematica's equivalent is In[99]:= TrigToExp[%97] Out[99]= -(\[Pi]/2) + I Log[-((7 I)/9) + (4 Sqrt[2])/9] - 2 I Log[I/3 + (2 Sqrt[2])/3] In[100]:= FullSimplify[%] Out[100]= 0 So it's pretty much a bug that FullSimplify forgot to try TrigToExp, or got scared by the intermediate swell. I'll bet I have somewhere some more egregious cases. Anyone else? --rwg On Sun, Sep 28, 2014 at 6:13 PM, Bill Gosper <billgosper@gmail.com> wrote: What is the "simplest" 0 FullSimplify can't see? Extra credit if FunctionExpand fails too. I'll start the bidding with In[97]:= FullSimplify[FunctionExpand[Subtract @@ %]] Out[97]= ArcCos[-(7/9)] - 2 ArcSec[3] In[98]:= TrigExpand[Cos[%]] Out[98]= 1 --rwg ________________________________