Selection of the numerical method 
Given the transcendental equation:
  $$ \frac{\tan x}{x} + c = 0, $$  where $c$ is any real number.

I tried Newton's method, but it is very bad fit. Which numerical method will be the smartest solution in this case?

If you can find an interval which contains a single root with a sign change, and no singularities, then the secant method or a variant on it should work. This method avoids the problems associated with picking a bad starting point. 
