Here is the answer to your query.
Let α be the semi vertical angle of a cone of slant height l
Then V O = l cos α and OA = l sin α
Let V be the volume of the cone Then
for maximum or minimum V, we must have
Differentially (1) w. r. t. α we get
The V is maximum when tan α = or α = tan–1
i.e. when the semi vertical angle of the cone is tan–1