I have in mind all the real roots of the equation 2*tan(Pi*t^2)-tan(Pi*t)+tan(Pi*t)*tan(Pi*t^2)^2 = 0.
Maple fails with it:
>RealDomain:-solve(2*tan(Pi*t^2)-tan(Pi*t)+tan(Pi*t)*tan(Pi*t^2)^2 = 0, t); RootOf(tan(_Z)*tan(_Z^2/Pi)^2-tan(_Z)+2*tan(_Z^2/Pi))/Pi
Even its numerical solution has gaps.
>Digits := 15; a := Student[Calculus1]:-Roots(2*tan(Pi*t^2)-tan(Pi*t)+tan(Pi*t)*tan(Pi*t^2)^2 = 0, t = -2 .. 2); Warning, some roots are returned as numeric approximations [-1.35078105935821, -1.18614066163451, -1.00000000000000, 0, 1.00000000000000, 1.28077640640442, 1.68614066163451, 1.85078105935821]>nops(a); 8 >b := Student[Calculus1]:-Roots(2*tan(Pi*t^2)-tan(Pi*t)+tan(Pi*t)*tan(Pi*t^2)^2 = 0, t = -2 .. 2, numeric); [-1.35078105935821, -1.18614066163451, -1.00000000000000, -0.780776406404415, 0., 1.00000000000000, 1.28077640640442, 1.68614066163451, 1.85078105935821, 2.00000000000000]>nops(b); 10
whereas
>plot(2*tan(Pi*t^2)-tan(Pi*t)+tan(Pi*t)*tan(Pi*t^2)^2, t = -2 .. 2);
shows 14 solutions.
The output of the command
>identify(a); [1/4-(1/4)*sqrt(41), 1/4-(1/4)*sqrt(33), -1, 0, 1, 1/4+(1/4)*sqrt(17), 1/4+(1/4)*sqrt(33), 1/4+(1/4)*sqrt(41)]
suggests a closed-form expression for the roots.