'Consider, *IF* an IR homing AAM is rated at 10 km effective engagement range in tail-chase, and that tail-chase range reduced to 1/4th from head-on engagement, that would mean the same missile could do 40 km head-on. '
That would depend on the relative speeds of the IR missile and tail-chased target. Assuming the speed of the IR missile doubles that of the tail-chased target, the head-on range would then be twice the tail-chased range i.e. 20 km. Similarly, if the relative speeds were 3 times, that would translate to a head-on range of 15 km.
In theory in kinetic terms this maybe true. In real life the IR AAM is limited in frontal engagement by the sensor detection and lock on range. Lock on is much more difficult in the front than it is in the back.