El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
$$\tan{\left(x \right)} - \cot{\left(\frac{x}{4} \right)} = 0$$
Resolvermos esta ecuaciónPuntos de cruce con el eje X:
Solución analítica$$x_{1} = - \frac{18 \pi}{5}$$
$$x_{2} = - 2 \pi$$
$$x_{3} = \frac{2 \pi}{5}$$
$$x_{4} = 2 \pi$$
$$x_{5} = - 4 i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}$$
$$x_{6} = - 4 i \log{\left(- \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}$$
$$x_{7} = - 4 i \log{\left(- \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}$$
$$x_{8} = - 4 i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} - \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}$$
$$x_{9} = - 4 i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}$$
$$x_{10} = - 4 i \log{\left(- \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{5} i}{4} + \frac{i}{4} \right)}$$
Solución numérica$$x_{1} = -21.3628300444106$$
$$x_{2} = 36.4424747816416$$
$$x_{3} = -69.1150383789755$$
$$x_{4} = 28.9026524130261$$
$$x_{5} = -96.7610537305656$$
$$x_{6} = -41.4690230273853$$
$$x_{7} = -61.5752160103599$$
$$x_{8} = 6.28318530717959$$
$$x_{9} = -64.0884901332318$$
$$x_{10} = -76.654860747591$$
$$x_{11} = -33.9292006587698$$
$$x_{12} = -81.6814089933346$$
$$x_{13} = 43.9822971502571$$
$$x_{14} = 69.1150383789755$$
$$x_{15} = -99.2743278534375$$
$$x_{16} = -94.2477796076938$$
$$x_{17} = 76.654860747591$$
$$x_{18} = -38.9557489045134$$
$$x_{19} = 3.76991118430775$$
$$x_{20} = -66.6017642561036$$
$$x_{21} = -74.1415866247191$$
$$x_{22} = 41.4690230273853$$
$$x_{23} = -71.6283125018473$$
$$x_{24} = -91.734505484822$$
$$x_{25} = 71.6283125018473$$
$$x_{26} = 81.6814089933346$$
$$x_{27} = -6.28318530717959$$
$$x_{28} = 96.7610537305656$$
$$x_{29} = -51.5221195188726$$
$$x_{30} = 23.8761041672824$$
$$x_{31} = 31.4159265358979$$
$$x_{32} = 94.2477796076938$$
$$x_{33} = 13.8230076757951$$
$$x_{34} = -54.0353936417444$$
$$x_{35} = -46.4955712731289$$
$$x_{36} = -11.3097335529233$$
$$x_{37} = 74.1415866247191$$
$$x_{38} = 21.3628300444106$$
$$x_{39} = 54.0353936417444$$
$$x_{40} = 18.8495559215388$$
$$x_{41} = -16.3362817986669$$
$$x_{42} = 84.1946831162065$$
$$x_{43} = -26.3893782901543$$
$$x_{44} = -23.8761041672824$$
$$x_{45} = -59.0619418874881$$
$$x_{46} = -79.1681348704628$$
$$x_{47} = 89.2212313619501$$
$$x_{48} = 8.79645943005142$$
$$x_{49} = 11.3097335529233$$
$$x_{50} = 99.2743278534375$$
$$x_{51} = -8.79645943005142$$
$$x_{52} = 46.4955712731289$$
$$x_{53} = 16.3362817986669$$
$$x_{54} = -56.5486677646163$$
$$x_{55} = -43.9822971502571$$
$$x_{56} = -86.7079572390783$$
$$x_{57} = -13.8230076757951$$
$$x_{58} = -18.8495559215388$$
$$x_{59} = -3.76991118430775$$
$$x_{60} = 33.9292006587698$$
$$x_{61} = 91.734505484822$$
$$x_{62} = 64.0884901332318$$
$$x_{63} = 61.5752160103599$$
$$x_{64} = -31.4159265358979$$
$$x_{65} = -28.9026524130261$$
$$x_{66} = 79.1681348704628$$
$$x_{67} = -84.1946831162065$$
$$x_{68} = 59.0619418874881$$
$$x_{69} = 66.6017642561036$$
$$x_{70} = 26.3893782901543$$
$$x_{71} = 56.5486677646163$$
$$x_{72} = -49.0088453960008$$
$$x_{73} = 38.9557489045134$$
$$x_{74} = 49.0088453960008$$