Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$\cos{\left(x \right)} - \frac{\cos{\left(2 x \right)}}{2} = 0$$
Solve this equationThe points of intersection with the axis X:
Analytical solution$$x_{1} = i \left(\log{\left(2 \right)} - \log{\left(- \sqrt{3} + 1 - \sqrt{2} \cdot \sqrt[4]{3} i \right)}\right)$$
$$x_{2} = i \left(\log{\left(2 \right)} - \log{\left(- \sqrt{3} + 1 + \sqrt{2} \cdot \sqrt[4]{3} i \right)}\right)$$
Numerical solution$$x_{1} = -77.3437544456587$$
$$x_{2} = 71.0605691384791$$
$$x_{3} = 42.0367663907535$$
$$x_{4} = 64.7773838312995$$
$$x_{5} = -79.735878233831$$
$$x_{6} = 48.3199516979331$$
$$x_{7} = -8.22871606668322$$
$$x_{8} = -20.7950866810424$$
$$x_{9} = -42.0367663907535$$
$$x_{10} = 54.6031370051126$$
$$x_{11} = -92.3022488481902$$
$$x_{12} = -130.001360691268$$
$$x_{13} = 20.7950866810424$$
$$x_{14} = -10.6208398548555$$
$$x_{15} = -73.4526929266514$$
$$x_{16} = -67.1695076194718$$
$$x_{17} = 4324.77702209906$$
$$x_{18} = 14.5119013738628$$
$$x_{19} = -89.9101250600178$$
$$x_{20} = 89.9101250600178$$
$$x_{21} = 73.4526929266514$$
$$x_{22} = -96.1933103671974$$
$$x_{23} = 98.5854341553698$$
$$x_{24} = -98.5854341553698$$
$$x_{25} = -14.5119013738628$$
$$x_{26} = -23.1872104692147$$
$$x_{27} = 39.6446426025812$$
$$x_{28} = -71.0605691384791$$
$$x_{29} = 10.6208398548555$$
$$x_{30} = -35.7535810835739$$
$$x_{31} = 8.22871606668322$$
$$x_{32} = 79.735878233831$$
$$x_{33} = -52.2110132169403$$
$$x_{34} = -29.4703957763943$$
$$x_{35} = 35.7535810835739$$
$$x_{36} = 92.3022488481902$$
$$x_{37} = 33.3614572954016$$
$$x_{38} = 1.94553075950364$$
$$x_{39} = 16.9040251620351$$
$$x_{40} = 29.4703957763943$$
$$x_{41} = -86.0190635410106$$
$$x_{42} = -39.6446426025812$$
$$x_{43} = 60.8863223122922$$
$$x_{44} = -48.3199516979331$$
$$x_{45} = 45.9278279097607$$
$$x_{46} = -45.9278279097607$$
$$x_{47} = 58.4941985241199$$
$$x_{48} = 96.1933103671974$$
$$x_{49} = 4.33765454767595$$
$$x_{50} = -27.078271988222$$
$$x_{51} = 77.3437544456587$$
$$x_{52} = -4.33765454767595$$
$$x_{53} = -58.4941985241199$$
$$x_{54} = -33.3614572954016$$
$$x_{55} = -1.94553075950364$$
$$x_{56} = 86.0190635410106$$
$$x_{57} = 52.2110132169403$$
$$x_{58} = -83.6269397528383$$
$$x_{59} = -54.6031370051126$$
$$x_{60} = 83.6269397528383$$
$$x_{61} = -60.8863223122922$$