How can I simplify this expression with trigonometric in Maple?

Question

cot(5*Pi*(1/22))+4*sin(2*Pi*(1/11))

can be simplify to sqrt(11),I tried to use simplify and combine, but I haven't been able to get that result with maple, why?

Answer 1

Simplification to a particular form or class of expression (say, involving only radicals and not trig) can sometimes be harder than zero detection. And Maple's simplify succeeds in determining that your input expression minus sqrt(11) is zero.

restart:
e := cot(5*Pi*(1/22))+4*sin(2*Pi*(1/11)):
simplify(e - sqrt(11));
                           0

But of course you may not know in advance any result only involving rationals and radicals, so doing only the above contains too much human intervention ("cheating"). However, there are sometimes a few ways in which you might "fairly" compute purely exact candidates for that zero testing, where the candidates involve radicals but not trig, say.

cand1 := sqrt(simplify(e^2));

                          (1/2)
                        11     

simplify( cand1 - e );
                           0

cand2 := identify(evalf(e));

                          (1/2)
                        11     

simplify( cand2 - e );
                           0