Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$\left(\frac{1}{2}\right)^{\tan{\left(x \right)}} \cot{\left(x \right)} = 0$$
Solve this equationThe points of intersection with the axis X:
Analytical solution$$x_{1} = \frac{\pi}{2}$$
Numerical solution$$x_{1} = -51.8498627602501$$
$$x_{2} = 36.1005956555676$$
$$x_{3} = 86.3662881672622$$
$$x_{4} = 42.3834308099153$$
$$x_{5} = 32.968497200569$$
$$x_{6} = 64.3748451632228$$
$$x_{7} = -1.59932716490056$$
$$x_{8} = -45.5820103997616$$
$$x_{9} = 10.9682228616175$$
$$x_{10} = 14.1090512740631$$
$$x_{11} = -23.590676893115$$
$$x_{12} = -29.8739291932351$$
$$x_{13} = -7.88245523671838$$
$$x_{14} = 20.3920411398722$$
$$x_{15} = 303.146486226473$$
$$x_{16} = 58.0921844492462$$
$$x_{17} = 80.0838257920694$$