Excellent question I must say (hehe I love these emoticons,cute little guys)
The answer is yes. Sunscreen has to be inevitably applied on cloudy and rainy days too.Its a common myth that sunscreen is not required on cloudy/rainy days but UV rays reach earth's surface on cloudy rays too.
The reason being is Ultraviolet rays can penetrate pretty much any surface. One might not necessarily get tanned but skin damage is possible.
One is susceptible to sun damage sitting inside tinted car windows,sitting near windows and standing under shade too. So a sunscreen is must!