If the Sun is a spotlight, daytime would be a circle and very small compared to night time. Yet this isn't true as about half of Earth is in daytime, actually more than half due to atmospheric scattering.
Look at this day and night world map
http://www.timeanddate.com/worldclock/sunearth.html it's impossible to be done by a spotlight in a flat disk. I draw a line to represent the terminator on the flat Earth map 42 mins ago and most of Antarctica is in daylight and the nighttime is smaller compared to the daytime.