We can play D Wilson's game with Chebyshevs because T_m ○T_n = T_mn . Are these just conjugates of x² - 2? Then why can't I guess the formula for the hyperbolish boundary gosper.org/chebcheb.png ? A possible approach to simplification: Even when it fails at a specific point, e.g., In[432]:= %370 /. x -> -1/Sqrt[5] // FullSimplify Out[432]= 1/10 (6 + (1/ 2 ((-1 - 2 I)^Sqrt[2] + (-1 + 2 I)^Sqrt[2] - . . . (where %370 is T_√2 ○T_√2(x) - 2 x² +1, as in the png), Series can sometimes cough up any number of 0 terms: In[438]:= Series[%370,{x,-1/Sqrt[5],4}]//tim During evaluation of In[438]:= Series::ztest: Unable to decide whether numeric quantities {3/5+1/2 (1/2 5^ . . . (-1+2 I)^Sqrt[2]]} are equal to zero. Assuming they are. During evaluation of In[438]:= 12.774193,6 Out[438]= O[x+1/Sqrt[5]]^5 But if you accept that the expression vanishes at specific points, you might believe that it does for a region peppered with such points. David's remark that there should be branch choices extending the validity of these identities must be true, because of analytic continuation, But it's not clear to me where to switch branches. --rwg gosper.org/cheb1.png shows T_2 ○T_½(x) - x with a point discontinuity at the origin, and T_½ ○T_2(x) - x with an isolated ray discontinuity along the canonical cut. Bruiser: Using the radicals for In[451]:= ChebyshevT[m, x] Out[452]= 1/2 ((x - Sqrt[-1 + x^2])^m + (x + Sqrt[-1 + x^2])^m) show T_mn = T_nm.