The tangent of the angle between the side edge of a regular hexagonal pyramid and the plane of its base is 32.

The tangent of the angle between the side edge of a regular hexagonal pyramid and the plane of its base is 32. Find the tangent of the angle between the plane of the side face and the plane of the base of the pyramid.

We have two triangles AOB and BOB, B the top of the pyramid, they have a common leg ОВ = tgОАВ * ОА = 32 * ОА; OA = OB / 32;
So the base of a regular hexagonal pyramid has angles = 120, angle OAN = 60, angle AOH = 30, AH = OA / 2 = OB / 64;
Cathet OH = √ (ОВ ^ 2/32 ^ 2-ОВ ^ 2/64 ^ 2) = ОВ * (√3) / 64; tgОАВ = ОВ / ОН = ОВ / ОВ * (√3) / 64 = 64 / (√3);
tangent of the angle between the plane of the side face and the plane of the base of the pyramid = 64 / (√3)