sin(7t) is periodic with period 2pi/7 while 2sin(10t) is periodic with period 2pi/10. The smallest number, which is a multiple of both periods, that is, of 2pi/7 and 2pi/10, is 2pi, which is then the period of the sum.

For the other problem, cos(2t) has period pi, while cos(2pi*t) has period 1. There is no number, which is an integer multiple of both of these periods, so the function cos(2t) + cos(2pi*t) is not periodic.