[math-fun] Before I make a fool of myself to Wolfram Support

Can anyone justify this? In[1467]:= Table[(Cot[(k \[Pi])/p] Sin[k \[Pi]])/p, {p, 4}] Out[1467]= {Cos[k \[Pi]], 1/2 Cot[(k \[Pi])/2] Sin[k \[Pi]], 1/3 Cot[(k \[Pi])/3] Sin[k \[Pi]], 1/4 Cot[(k \[Pi])/4] Sin[k \[Pi]]} In[1468]:= FullSimplify[%, k \[Element] Integers] Out[1468]= {(-1)^k, Cos[(k \[Pi])/2]^2, 0, 0} Millions(?) of users, dozens of years, yet things like In[1471]:= FullSimplify[{I^k - I^(5*k) + 1, I^k - I^(5*k) + one}, k \[Element] Integers] Out[1471]= {1 + I^k - I^(5 k), one} Sometimes it feels like reality rot. --rwg