In order to change the rotation of the single phase induction motor, the connection to one of the stator windings(starting winding and main winding) is reversed. But later we cut the starting winding ,so how is it possible to reverse the direction of rotation of the single phase induction winding?
You have to stop the motor before you can reverse it.
That lets the starter winding establish the new direction (and if the starter winding is disconnected by a centrifugal switch, it lets the switch turn on again.
Capacitor and split phase start induction motors don't need the start winding to run.
They need it only to get them started to one direction. The start winding has a phase shift against the run winding. Without the start winding the motor will hum and burn, unless it gets the momentum some other way.
It is very dangerous to spin a single phase motor that is plugged in and not running. Once it gets direction, it starts immediately.
Similar is a 3 phase motor that loses one phase, will continue to run as a single phase motor until stopped.