Just use the default events of these controls to have one update the other. Double-click the controls to add the event handlers and make them look like this:
private void dateTimePicker1_ValueChanged(object sender, EventArgs e) {
monthCalendar1.SetDate(dateTimePicker1.Value);
}
private void monthCalendar1_DateChanged(object sender, DateRangeEventArgs e) {
dateTimePicker1.Value = monthCalendar1.SelectionStart;
}
Do note that DateTimePicker already has a MonthCalendar built-in, you get it when you click the dropdown button. Which is the typical way it is used, they are already in sync.