It sounds like you want register_taxonomy_for_object_type to add the taxonomy to the “Page” post type.
register_taxonomy_for_object_type('events-cat','page');
As noted, I don’t know how you expect this to work. You are going to have to alter the page templates.
I would also suggest that you look into the Template Hierarchy, in particular the part about custom taxonomies. It sounds like you may be reinventing Core functionality.