As Whopity says plus:
A conformal conic projection is not exactly a conic projection, i.e. the projection of a sphere onto a cone. The conformal condition (i.e. the same scale locally in any direction) requires the solution of a partial differential equation, which is definitely beyond O-level maths.
It is not possible to model the surface of a sphere exactly onto any flat surface (which includes cones because you can unwrap them). So arcs of arbitrary Great Circles will not, in general, be straight lines in any chart projection, though they can be "straight enough" within the confines of the chart, as Whopity said.