spherical geometry - Finding the northernmost latitude in a great circle that passes through two points on the sphere

I think it is a little more straightforward if you work in Cartesian coordinates.

Take an earth of radius one, for simplicity.

Then we can take the longitudes as $pm l$ without loss of generality. Then
the halfway point between the two points, projected to the surface of the earth, will have the highest latitude (equivalently the highest $z$ component).

The two points are $(cos phi cos l, cos phi sin l, sin phi)$ and
$(cos phi cos l, -cos phi sin l, sin phi)$, and the mid point is
$(cos phi cos l, 0, sin phi)$. To project to the surface, we divide by
the norm, to get a $z$ component (on our unit earth) of
$sin delta = {sin phi over sqrt{ (cos phi cos l)^2 + sin^2 phi}} = { tan phi over sqrt{cos^2l+tan^2 phi} }$.

To obtain the $arctan$, we first need $cos delta$, which we get from
$cos delta = sqrt{1 -sin^2 delta} = { cos l over sqrt{ cos^2l+tan^2 phi} } $, and hence
$tan delta = {tan phi over cos l}$, which is the desired result.